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A TKEATISE 



LEVELLING, TOPOGRAPHY AND 
HIGHER SURVEYING. 



BY 

"W. M. GILLESPIE, LL.D.. 



if 

CIVIL ENGINEER. 



EDITED BY 



CADY STALEY, A. M., 0. E. 



NEW YOEK: 
D. APPLETON AND COMPAM, 

549 & 551 BROADWAY. 

1876. 



.G-1-7 



Entered, according to Act of Congress, in the y^ar 1870, 

By D. APPLETON & COMPANY, 

In the Offi.ce of the Librarian of Congress, at Washington. 



By Exchange 
Army and Navy Club 

JANUARY 16 1934 



i3~MifZ 



C^ 






PEEFACE. 



This volume, announced several years ago as being in 
preparation to follow the author's Land Surveying, was left 
at the time of his death unfinished. In preparing it for the 
press, the editor has endeavored to carry out, as far as pos- 
sible, the plan already laid down. A considerable part of the 
volume has been given by the author, in the form of lectures 
to the civil engineering classes in Union College, and has 
been printed from the original manuscript. 

The best authors on the subjects treated have been con- 
sulted, in order to render the work as complete as possible. 
The principal anthorities are the following: Begat, Bourns, 
Breton, Chauvenet, Fen wick, Frome, Gurley, Guyot, Jackson 
Lee, Narrien, Puissant, Salneuve, Simms, Smith, Stephenson, 
Wiesbach, Williams, and the papers of the United States 
Coast Survey. 

Wherever it has been necessary to refer to the elementary 
principles of surveying, reference has been made to Gillespie's 
Land Surveying (designated as L. S. for brevity), and the 
numbers of the articles referred to are enclosed in parentheses. 

Union College, 
Schenectady, June, 1870. 



ANALYTICAL TABLE OF CONTENTS. 



INTRODUCTION. 

ART. PAGB 

1. Levelling in General 1 

2. Direct Levelling 1 

3. Indirect Levelling 2 

4. Barometric Levelling 2 

5. Topography. 2 

G. Special Objects and Difficulties 3 

7. Underground Surveying 3 

8. Water Surveying 3 

9. Reflecting Instruments 3 

10. Spherical Surveying 3 

11. Location 4 



PART I. 

DIRECT LEVELLING. 
CHAPTER I. GENERAL PRINCIPLES. 

12. Levelling Instruments 5 

13. Methods of Operation 5 

14. Curvature 6 

15. Refraction 7 

CHAPTER II. PERPENDICULAR LEVELS. 

16. Principle 7 

17. Plumb-line Levels 7 

18. Reflecting Levels 8 



vi ANALYTICAL TABLE OF CONTENTS. 



CHAPTER III. WATER-LEVELS. 

AKT. PAGE 

19. Continuous Water-levels 10 

20. Visual Water-levels 11 



CHAPTER IV. AIR BUBBLE OR SPIRIT LEVELS. 

21. The Spirit-level 12 

22. Sensibility 12 

23. Block-level 13 

24. Level with Sights ,14 

25. Hand Eeflected Level 14 

26. The Telescope Level. 15 

27. The Y Level 16 

28. The Telescope 16 

29. The Cross-hairs 17 

30. The Level 17 

31. Supports 18 

32. Parallel Plates 18 

33. Adjustments 18 

34. First Adjustment • .19 

35. Second Adjustment • 20 

36. Third Adjustment 20 

37. Centring the Object-glass and Eye-piece 21 

38. Adjustment by the " Peg Method " 22 

39. Verification by another Telescope 23 

40. Egault's Level 23 

41. Troughton's Level 23 

42. Gravatt's Level '. 24 

43. Bonrdaloue's Level 24 

44. Lenoir's Level ' 24 

45. Tripods 25 



CHAPTER V. RODS. 

46. How made 25 

47. Target : 26 

48. Vernier's 27 

49. New-York Rod 27 

50. Boston Rod 27 

51. Speaking Rods 28 



ANALYTICAL TABLE OF CONTENTS. Y i{ 



CHAPTER VI. THE PRACTICE. 

52. Field Eoutine 30 

53. Field-notes 32 

54. First Form of Field-book 33 

55. Second Form of Field-book 35 

56. Third Form of Field-book ' 38 

57. Best Length of Sight 39 

58. Equal Distances of Sight 39 

59. Datum-level 39 

60. Bench-marks 40 

61. Test-levels 40 

62. Limits of Precision ; 41 

63. Flying-levels 41 

64. Levelling for Sections 41 

65. Profiles , 41 

66. Gross-levels 42 



CHAPTER VII. — DIFFICULTIES. 

67. Steep Slopes . . ; 43 

68. "When the Eod is too low 44 

69. When the Eod is too high 45 

70. When the Eod is too near 1 ........ . 45 

71. Levelling across Water 45 

72. Across a Swamp or Marsh 40 

73. Through Underwood 46 

74. Over a Board Fence 46 

75. Over a Wall 46 

76. Through a House ....... 47 

77. The Sun ' 47 

78. Wind 48 

79. Idiosyncrasies : 48 

80. Eeciprocal Levelling 48 



CHAPTER Vin. LEVELLING LOCATION. 

81. Its Nature 49 

82. Difficulties '. 49 

83. Staking out Work 50 

84. To Locate a Level Line 51 

85. Applications 51 

86. To run a Grade Line 52 



Viii ANALYTICAL TABLE OF CONTENTS. 

PART II. 

INDIRECT LEVELLING. 
CHAPTER I. METHODS AND INSTRUMENTS. 

ART. PAGE. 

87. Vertical Surveying 53 

88. Vertical Angles 54 

89. Instruments 55 

90. Slopes 56 

91. Theodolites 56 

92. Surveyor's Transit 56 

93. Adjustments 58 

94. Field-Work 60 

95. Angular Profiles 61 

96. Buruier's Level 62 

97. German Universal Instrument 62 

CHAPTER n. SIMPLE ANGULAR LEVELLING. 

A. — For Short Distances. 

98. Principle 63 

99. Best-conditioned Angle i 63 

B. — For Greater Distances. 

100. Correction for Curvature 64 

101. Correcting the Result 64 

102. Correcting the Angle „ 64 

103. Correction for Refraction 64 

C. — For Very Great Distances. 

104. Correction for Curvature 05 

105. Correction for Refraction QQ 

106. Reciprocal Observation for cancelling Refraction '. 67 

107. Reduction to the Summits of the Signals 67 

108. When the Height of the Signal cannot be Measured 68 

109. Levelling by the Horizon of the Sea 69 

CHAPTER ni. COMPOUND ANGULAR LEVELLING. 

110. By Angular Coordinates in one Plane. 70 

111. By Angular Coordinates in several Planes : . . 71 

112. Conversely 71 



ANALYTICAL TABLE OF CONTENTS. j x 

PART III. 

BAROMETRIC LEVELLING. 

CHAPTER I. PRINCIPLES AND FORMULAS. 

AKT. PAGE 

113. Principles 73 

114. Applications 73 

115. Correction for Temperature of the Mercury 74 

116. Correction for Temperature of the Air 74 

117. Other Corrections 74 

118. Eules for calculating Heights 75 

119. Formulas . . ; 75 

120. To Correct for Latitude ' 76 

121. Final English Formula 76 

122. French Formulas 77 

123. Babinet's Formula 77 

124. Tables 78 

125. Approximations 78 

CHAPTER n. INSTRUMENTS. 

126. Mountain Barometers 79 

127. The Aneroid Barometer '. 80 

128. " Boiling-point Barometer " 80 

129. Accuracy of Barometric Observations 81 

130. Simultaneous Observations 81 



PART IY. 

TOPO GRAPHT. 
INTRODUCTION. 



131. Definition 

132. Systems . . 



CHAPTER I. BY HORIZONTAL CONTOUR-LINES. 

133. General Ideas 83 

134. Plane of Reference 84 

135. Vertical Distances of the Horizontal Sections. . . ; 84 

136. Methods for determining Contour-lines 84 



fc ANALYTICAL TABLE OF CONTENTS. , 

First Method. 

^ ET - PAGE 

137. General Method - 84 

138. On a Long, Narrow Strip 85 

139. On a Broad Surface ' 85 

140. Surveying the Contour-lines 85 

141. Contouring with the Plane-table 86 

Second Method. 

142. General Nature 86 

143. Irregular Ground 86 

144. On a Single Hill : 87 

145. Extensive Topographical Survey 87 

146. Interpolation 88 

147. Interpolating with the Sector 88 

148. Eidges and Thalwegs 89 

149. Forms of Ground 90 

150. Sketching Ground by Contours 91 

151. Ambiguity 91 

152. Conventionalities 92 

153. Applications of Contour-lines 92 

154. Sections by Oblique Planes 92 

CHAPTER II. BY LINES OF GREATEST SLOPE. 

155. Their Direction 93 

156. Sketching Ground by this System 93 

157. Details of Hatchings 93 

CHAPTER III. SHADES FROM OBLIQUE AND VERTICAL LIGHT. 

158. Degree of Shade 94 

159. Shades by Tints 94 

160. Shades by Contour-lines 95 

161. Shades by Lines of Greatest Slope 95 

162. The French Method 95 

163. Lehmann's Method 95 

164. Diapason of Tints 97 

165. Shades Produced by Oblique Light 97 

CHAPTER IV. CONVENTIONAL SIGNS. 

166. Signs for Natural Surface 98 

167. Signs for Vegetation „ 98 

168. Signs for Water 99 



ANALYTICAL TABLE OF CONTENTS. x i 

ABT. PAGE 

169. Colored Topography 100 

170. Signs for Miscellaneous Objects 101 

171. Scales 103 



PART Y. 



UND ERGRO UND R MININ G SURVE YING . 

172. Objects , 105 

CHAPTER I, SURVEYING AND LEVELLING OLD LINES. 

173. First Object 105 

174. The Old Method , 106 

175. The New Method 108 

176. The Mining Transit 109 

177. Mapping. 109 

CHAPTER II. LOCATING NEW LINES. 

178. Second Object 110 

179. When the Mine is entered by an Adit 110 

180. When the Mine is entered by a Shaft Ill 

181. To Dispense with the Magnetic Needle Ill 

182. Eeducing Several Courses to One 112 

183. Third Object 113 

184. Problems 113 



PART VI. 

THE SEXTANT AND OTHER REFLECTING INSTRUMENTS. 

CHAPTER I. — THE INSTRUMENTS. 

185. Principle 115 

186. Description of the Sextant 117 

187. The Box Sextant 118 

188. The Reflecting Circle 118 

189. Adjustments of the Sextant 118 

190. How to Observe 120 

191. Parallax of the Sextant. . . 120 



x ii ANALYTICAL TABLE OF CONTENTS. 
CHAPTER II. THE PRACTICE. 

AKT. PAGE 

192. To Set Out Perpendiculars 121 

193. The Optical Square 121 

194. To Measure a Line, one end being inaccessible 122 

195. Otherwise 124 

196. To Measure a Line when both ends are inaccessible 124 

197. Obstacles 124 

198. To Measure Heights 124 

199. Artificial Horizon 125 

200. The Sun 126 

201. Very Small Altitudes and Depressions 126 

202. To Measure Slopes 127 

203. Oblique Angles 128 

204. Advantages of the Sextant 129 



PART VII. 

MARITIME OB ETD BO GRAPHICAL STTBYEYING. 

205. Object 131 

CHAPTER I. THE SHORE LINE. 

206. The High-water Line 131 

207. The Low-water Line 132 

208. Measuring the Base 132 

CHAPTER H. SOUNDINGS. 

209. In Farrow Water 133 

210. Finding the Position of a Boat on a Sea-coast 134 

211. From the Shore , 134 

212. From the Boat, with a Compass 134 

213. From the Boat, with a Sextant 134 

214. Trilinear Surveying 135 

215. Problem of the Three Points. 135 

216. Instrumental Solution 137 

217. Analytical Solution : 137 

218. Between Stations 138 

219. The Sounding-line ! 139 



ANALYTICAL TABLE OF CONTENTS. x iii 

CHAPTER m. TIDE-WATERS. 

ABT. PAGE 

220. Tides '. 140 

221. Difference on Atlantic and Pacific Coast 140 

222. Mean Level of the Sea. 141 

223. High and Low Water 141 

224. " Establishment " of a Place 141 

225. Tide Ganges 141 

226. Tide Tables ' 142 

227. Gauges in Bends 144 

228. Beacons and Buoys 144 



CHAPTER VI. THE CHART. 

229. Methods of Fixing Points on the Chart 145 

230. Conventional Signs 146 



PART VIII. 

SPHERICAL SURVEYING, OR GEODESY. 
CHAPTER I. THE FIELD-WORK. 

231. Nature 147 

232. Triangular Surveying , 147 

233. Outline of Operations 148 

234. Measuring the Base 148 

235. Corrections of the Base ..'. 152 

236. Reducing the Base to the Level of the Sea 152 

237. A Broken Base 153 

238. Base of Verification 153 

239. Choice of Stations 154 

240. Signals , 157 

241. Observations of the Angles 159 

242. Reduction to the Centre 160 

243. The Angles 162 

244. " Spherical Excess " 162 

245. Correction of the Angles 164 

246. Interior Eilling-up • 164 



x { v ANALYTICAL TABLE OF CONTENTS. 

CHAPTER II. — CALCULATING THE SIDES OF THE TRIANGLES. 

ART. PAGE 

247. Methods - 165 

248. Delambre's Method 165 

249. Legendre's Method ; 166 

250. Coordinates of the Points 166 

251. Problem I 167 

252. Second Solution 168 

253. Problem IT 168 

254. Lee's Formulas 170 



LEVELLING, TOPOGRAPHY, AND HIGHER 
SURVEYING. 



INTRODUCTION. 

(1.) Levelling in General. A level surface is one which is 
everywhere perpendicular to the direction of gravity, as indi- 
cated by a plumb-line, etc. ; and, consequently, parallel to the 
surface of standing water. It is, therefore, spherical (more 
precisely, spheroidal), but, for a small extent, may be consid- 
ered as plane. Any line lying in it is a level line. 

A vertical line is one which coincides with the direction 
of gravity. 

The height of a point is its distance from a given level 
surface, measured perpendicularly to that surface, and there- 
fore in a vertical line. 

Levelling is the art of determining the difference of the 
heights of two or more points. 

To obtain a level surface or line, usually the latter, is the 
first thing required in levelling. 

When this has been obtained, by any of the methods to be 
hereafter described, the desired height of a point may be de- 
termined directly or indirectly. 

(2.) Direct Levelling. In this method of levelling, a level 
line is so directed and prolonged, either actually or visually, 
as to pass exactly over or under the point in question (i. e., 
so as to be in the same vertical plane with it), and the height 



2 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(or depth) of the point above (or below) this level line is 
measured by a vertical rod, or by some similar means. The 
height of any other point being determined in the same man- 
ner, the difference of the two will be the height of one of the 
points above the other. So on, for any number of points. 

Direct Levelling is the method most commonly em- 
ployed. It will form Part I. of this volume. 

(3.) Indirect Levelling. In this method of levelling the de- 
sired height is obtained by calculation from certain coordinate 
measured lines or angles, which fix the place of the point. 

Thus, the horizontal distance from any point to a tree 
being known, and also the angle with the horizon made by a 
straight line passing from the point to the top of the tree, its 
height above the point can be readily calculated. This is the 
most simple and most usual form of this method, though 
many others may be employed. 

Indirect Levelling will be developed in Part II. 

(4.) Barometric Levelling. This determines the difference of 
the heights of two points by the difference of the weights of 
the portions of the atmosphere which are above each of them ; 
as indicated by a barometer. It is explained in Part III. 

(5.) Topography. " Surveying " determines the position of 
one point with reference to another, supposing them both to 
be situated in (or reduced to) the same level plane. " Level- 
ling " determines how much the point in question is above or 
below some other level plane. Both of these combined de- 
termine where the point is " in space ; " that is, where it is in 
reference to some known point; both horizontally, i. e.^how 
far it is in front or behind, to right or to left, etc. ; and ver- 
tically, i. e., how far it is above or below. 

The position of a point in its own level plane is usually 
determined in " Surveying " by a pair of coordinates— lines 
or angles. [See L. S., (2), etc.] l Then, its vertical distance 

1 L. S. will for brevity, be used to denote the Author's " Treatise on Land 
Surveying," and the No. of the article referred to will be enclosed in ( ). 



INTRODUCTION. 3 

above or below a known level plane (i. e., its height or depth) 
being determined by " Levelling," becomes a " third coordi- 
nate," which fixes the place of the point. 

The application of snch combinations of Surveying and 
Levelling to determine the positions, in horizontal projection, 
and also the heights of the inequalities of a limited portion 
of the surface of the earth (its hills and hollows, ridges and 
valleys, etc.), is called Topography. Topographical Mapping 
represents these inequalities on paper. Topography on a 
larger scale becomes Geography, properly so called. Topog- 
raphy occupies Part IV. 

(6.) Special Objects and Difficulties. The preceding methods 
are sufficient for the complete determination of all the features 
of the earth's surface; but certain operations in particular 
places require special methods. 

(7.) Operations beneath the surface (for tunnelling, mining, 
etc.) being in darkness, and not easily connected with the 
above-ground work, involve some novel problems, and will, 
therefore, be treated separately in Part V., as Underground 
Surveying and Levelling. 

(8.) So too, operations on the water, because of the want of 
steadiness in positions on its surface, require peculiar methods, 
and constitute another modification ; described in Part YIL, 
as Water Surveying and Levelling. 

(9.) Reflecting Instruments, such as the sextant, being 
chiefly used in the above situation, are treated of in the pre- 
ceding Part YI. 

(10.) Spherical Surveying. When a great extent ol country 
is comprised in a survey, the surface of the earth can no longer 
be considered as plane, but its curvature must be taken into 
account. Then Spherical Surveying, or Geodesy, must be 
employed ; and, instead of the straight lines and plane angles 



4 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

which are the coordinates of Plane Surveying, arcs of circles 
and spherical angles mnst be used. It forms Fart Till. 

When still greater extents are to be surveyed, the methods 
of spherical surveying must be modified in accordance with 
the true spheroidal form of the earth. 

(11.) Location. The name Surveying is often made to in- 
clude a mode of operation which is precisely its converse. 

Surveying, properly so called, determines and records the 
relative positions of points as they really are. 

The converse operations have for their objects to fix the 
places of points where they are desired to be. 

The term Location may be extended beyond its usual 
meaning so as to embrace all such operations. 

In laying out land, parting off portions of it, and dividing 
it up, the desired lines are not surveyed, but located. 

In the United States Public Land Surveying, the work is 
almost entirely Location. 

The determination of the lines of roads, their curves, etc., 
is especially Location. 

The finding and pursuing a given course at sea (in Navi- 
gation) is only another form of it. 

We shall find many applications of this distinction be- 
tween Surveying and Location. A similar one occurs in 
Levelling. It should be carefully kept in mind both in " Sur- 
veying " and in " Levelling." 



PART I 



DIRECT LEVELLING 



CHAPTEE I. 



GENERAL PRINCIPLES 



(12.) Levelling Instruments. The instruments employed to 
obtain a level line may be arranged in three classes, depending 
on these three principles : 

1. That a line perpendicular to a vertical line is a hori- 
zontal or level line. 

2. That the surface of a liquid in repose is horizontal. 

3. That a bubble of air, confined in a vessel otherwise full 
of a liquid, will rise to the highest point of that liquid. 

They will be described in the following three chapters. 

(13.) Methods of Operation. When a level line has been ob- 
tained, by any means, the difference of heights of any two 
points may be found by either of these two methods : 

Fig. 1. 




First Method. — Set the levelling instrument over one of 
the points, as A, in Fig. 1 



Measure the height of the level 



6 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

line above the point. Then direct this line to a rod held on 
the other point, and note the reading. The difference of the 
two measurements at A and B will be the difference of their 
heights. 

Second Method. Let A and B, Fig. 2, represent the two 
points. Set the instrument on any spot from which both the 

points can be seen, and 

Fig. 2. , . 

at such a height that 
the level line will pass 
above the highest one. 
Sight to a rod held at 
A, and note the read- 
ing. Then turn the 
instrument toward B, 
and note the height ob- 
served on the rod held 
at that point. The difference of the two readings will be the 
difference of the heights required. The absolute height of the 
level line itself is a matter of indifference. 




(14.) Curvature. The level line given by an instrument is 
tangent to the surface of the earth. Therefore, the line of 
true level is always below the line of apparent level. In Fig. 3, 
A D represents the line of apparent level, 
and A B the line of true level. D B is 
the correction for the earth's curvature. 
By geometry we have : 

AD 2 = DB x (DB + 2BO). 

But D B, being very small, compared with 

the diameter of the earth, may be dropped 

from the quantity in the parenthesis, and 

we have : 

AD 2 




DB = 



2BO 



i. e., the correction equals the square of 
the distance divided by the diameter of the earth. 



PERPENDICULAR LEVELS. 
The difference of height for a distance of 



1 mile = 



5280 x 12 



7916 



7916 



= 8 inches. 



This varies as the square of the distance. The effect, if 
neglected, is to make distant objects appear lower than they 
really are. 

The effect is destroyed by setting the instrument midway 
between the two points. 

(15.) Refraction. Rays of light coming through the air are 
curved downward. The effect is, to make objects look higher 
than they really are. Its amount is about \ that of curvature, 
and it operates in a contrary direction. 



CHAPTER II. 

PERPENDICULAR LEVELS. 

(16.) Principle. The principle upon which these are con- 
structed is, that a line perpendicular to the direction of gravity 
is a level line. 



(17.) Plumb-line Levels. The A level, Fig. 4, is so adjusted 
that when the plumb-line coincides with the mark on the 



Fro. 4. 



Fig. 5. 




ill 



cross-piece, the feet of the level shall be at the same height. 
It is adjusted by reversion thus : Place its feet on any two 
points. Mark on the cross-bar the place of the plumb-line. 



8 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



Turn the instrument end for end, resting it on the same 
points,- and mark the new place of the plumb-line. The point 
midway between the two is the right one. 

Another form is shown in Fig. 5. 

The above forms are not convenient for prolonging a level 
line. To do this, invert the preceding form, as in Fig. 6. 

Fig. 6. 



W" 



ill 

i 

To test and adjust this, sight to some distant point nearly 
on a level, and mark where the plumb-line comes to on the 
bottom of the rod. Turn the instru- 
ment around and sight again, and note 
the place of the plumb-line. The mid- 
way point is the right one. 

A modification of the last form is to 
fasten a common carpenter's square in 
a slit in the top of a staff, by means of 
a screw, and then tie a plumb-line at 
the angle so that it may hang beside 
one arm. When it has been brought to 
to do so, by turning the square, then the other arm will be 
level. 

(18.) Reflecting Levels. In these, the perpendicular to the 
direction of gravity is not an actual line, but an imaginary 
reflected line. 

It depends on the optical principle that a ray of light 
which meets a reflecting plane at right angles is reflected back 
in the same line. 

When the eye sees itself in a plane mirror, the imaginary 
line which passes from the eye to its image is perpendicular 
to the mirror. Therefore, if the mirror be vertical, the line 




PERPENDICULAR LEVELS. 



Fig. 8. 




will be horizontal. It may therefore be used like any other 
line of sight for determining points at the same height as itself. 

The first form, Fig. 8 (Colonel BureFs), consists of a rhomb 
of lead, of about 2 inches on a side, and 1 inch thick. 
■ One side (the shaded part of the figure) is faced with a 
mirror. The right-hand corner of the rhomb 
is cut off, as seen in the figure, and a wire, A B 
is stretched across the mirror. 

To use this, hold up the instrument, with 
the mirror opposite the eye, by the string D, A, 
so that the eye seems bisected in the mirror 
by the wire A B. Then glance through the 
opening at B, and any point in the line of the 
eye and wire will be in the same horizontal 
plane with them. 

The correctness of the instrument may be verified in the 
following manner : Hold up the instrument before any plane 
surface, as a wall, and determine the height of some point, as 
previously directed. Then, without changing the height of 
the instrument, turn it half around, place yourself between it 
and the wall, and note the point of the wall which is seen in 
the mirror to coincide with the image of the eye. 

If the two points on the wall coincide, the instrument is 
correct. If they do not, the mirror does not hang plumb, and 
the point midway between the two is the true one. 

The instrument is 
rectified, or made to 
hang plumb, by means 
of the pear - shaped 
piece of lead seen at- 
tached to the lower 
corner of the rhomb. 

The second form 
consists of a hollow 
brass cylinder, with an 
opening at the upper end, as seen in Fig. 9. At the opening 
is a small mirror, whose vertical plane makes an angle with 



Fig. 9. 





10 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



Fig 10. 



the vertical plane of section by which the cylinder was cnt in 
forming the aperture. The edge of the mirror is marked thus 
(x) in the first half of Fig. 9. The mirror is made to hang 
plumb by means of a one-sided weight within the cylinder. 

This is used by setting it on a stake driven into the ground, 

or by holding it in the hand, making the lower edge of the 

opening answer the same purpose as the wire in the other case. 

The same methods of verification and rectification are used 

as with the first form of the instrument. 

The instrument, in its third 
form, is simply a small steel 
cylinder, 4" or 5" long, and y 
in diameter, highly polished, 
- and suspended from the centre 
of one end by a fine thread. 

To use this, hold it up by the 
thread with one hand, and with 
the other hand hold a card be- 
tween the eye and instrument, 
using the upper edge of the card, as seen reflected in the 
mirror, the same as the wire in the first form. 

This instrument is the invention of M. Cousinery. 



**• 



"03 



\J 



■+•+- 



CHAPTEE III. 



WATER-LEVELS 



(19.) Continuous Water-levels. These may consist of a chan- 
nel connecting the two points, and filled with water ; or of a 
tube, usually flexible, with the ends turned up and extending 
from one point to the other. 

By measuring up or down, from the surface of the water at 
each end, the relative heights of the two points may be de- 
termined. 



WATEK-LEYELS. H 

(20.) Visual Water-levels. The simplest one is a short sur- 
face of water prolonged by sights at equal distances above it, 
as in Fig. 11. 

Fig. 11. 



A portable form is a tube bent up at each end, and nearly 
filled with water. The surface of the water in one end will 
always be at the same height as that in the other, however 
the position of the tube may vary. It may be easily con- 
structed with a tube of tin, lead, copper, etc., by bending up, 
at right angles, an inch or 



r\ 



fo 



two of each end, and sup- ^ 

porting the tube, if too ~ £ 

flexible, on a wooden bar. 

In these ends, cement (with 

putty, twine dipped in 

white-lead, etc.) thin phials, with their bottoms broken off, 

so as to leave a free communication between them. Fill the 

tube and the phials, nearly to their top, with colored water. 

Blue vitriol or cochineal may be used for coloring it. Cork 

their mouths, and fit the instrument, by a steady but flexible 

joint, to a tripod. 

To use it, set it in the desired spot, place the tube by eye 
nearly level, remove the corks, and the surfaces of the water 
in the two phials will come to the same level. Stand about 
a yard behind the nearest phial, and let one eye, the other 
being closed, glance along the right-hand side of one phial, 
and the left-hand side of the other. Raise or lower the head 
till the two surfaces seem to coincide, and this line of sight, 
prolonged, will give the level line desired. Sights of equal 
height, floating on the water, and rising above the tops of the 
phials, would give a better-defined line. 



12 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



CHAPTEE IV. 

AIR-BUBBLE OR SPIRIT LEVELS. 

(21.) The " Spirit-level " consists essentially ot a curved 

glass tube nearly filled with alcohol, but with a bubble of air 

PlG 13 left within, which always seeks 

^fr_j_ ■._■._• i __ the highest spot in the tube, 

~ r 1 1 C^Z-^=^ZZZr An \ an( i wili therefore, by its move- 

1 n ' ments, indicate any change in 

the position of the tube. When- 
ever the bubble, by raising or lowering one end, has been 
brought to stand between two marks on the tube, or, in case 
of expansion or contraction, to extend an equal distance on 
either side of them, the bottom of the block (if the tube be in 
one), or sights at each end of the tube, previously properly 
adjusted, will be on the same level line. It may be placed 
on a board fixed to the top of a staff or tripod. 

When, instead of the sights, a telescope is made parallel 
to the level, and various contrivances to increase its delicacy 
and accuracy are added, the instrument becomes the engi- 
neer's spirit-level. 

The upper surface of the tube is usually the arc of a circle, 
and when we speak of lines parallel to a " level," we mean 
parallel to the tangent of this arc at it's highest point, as indi- 
cated by the middle of the bubble. 

(22.) Sensibility. This is estimated by the distance which 
the bubble moves for any change of inclination. It is directly 
proportional to the radius of curvature of the tube. To de- 
termine the radius, proceed thus : 

Let S = length of the arc over which the bubble moves for 
an inclination of 1 second (1'). 



AIR-BUBBLE OR SPIRIT LEVELS. 



13 



Let K 



its radius of curvature. 

Then S : 2ttE : : 1": 360°, 
whence E = 206265 x S, 
E 



or S 



206265 



Fig. 14. 




S may be found by trial, the level being attached to a 
finely-divided vertical cir- 
cle. The radius may also 
be found without this, 
thus : Bring the bubble to 
centre, and sight to a di- 
vided rod. Eaise or lower 
one end of the level, and 
again sight to rod. Call 
the difference of the read- 
ings A, the distance of the 
rod d, and the space which 

the bubble moved S. Then we have two approximately sim- 
ilar triangles ; whence r=~jr- m 

Example. At 100 feet distance, the difference of readings 

was 0.02 foot, and the bubble moved 0.01 foot. Then the^ ra- 

100 x 0.01 ^ 
dius was — — — = 50 it. 

The sensibility of an air-bubble level equals that of a 
plumb-line level having a plumb-line of the same length as 
the radius of curvature. 



(23.) Block-level. If this is marked by the maker, and 
the bubble does not come to the 
centre, when turned end for end, 
plane or grind off one end of the 
bottom until it does. 

Otherwise, if the bubble-tube is capable of movement, 
raise or lower one end of it until it will verify, bringing the 



14 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

bubble half-way back to the middle by this means, and the 
other half by raising or lowering one end of the block ; be- 
cause the reversion has doubled the error. 
Eepeat this, if necessary. 

Circular Level. The upper surface of this is spherical. 
It will therefore indicate a level in every, di- 
rection, instead of only one, as does the pre- 
ceding. It is adjusted like the last one, but 
in two directions, at right angles to each 
other. 

(24.) Level with Sights. The line of sight is made parallel 
to the tangent of the level. It may be tested thus : 



Fig. 16. 




m 



Fig. 17. 



Bring the bubble to the centre of the tube and make a 
mark, in the line of sight, as far off as can be seen. Then 
turn the level end for end, and sight again. If the bubble 
remains in the same place, " all right." If not, rectify it by 
altering the sights, or by altering the marks for the bubble to 
come to, bringing the bubble half-way back, and trying it 
as-ain. 



(25.)^ Hand Reflected Level. This consists of a brass tube, 
about six inches long, and one inch in diameter. To the 
inside of the upper portion of the tube is attached a small 
level. A small mirror is placed at an angle in the lower side 
of the tube, so that it will reflect the point to which the bub- 
ble must come, in order to have the instrument level, to the 
eye. A small hole at one end, and a horizontal cross-hair at 



AIR-BUBBLE OR SPIRIT LEYELS. 



15 



the other, gives the desired level line. It is used by holding 
it in the hand. 



Fig. 18. 




Fig. 18 is an approved form, made by Young, of Phila- 
delphia. The improvement consists in the patent "Locke 
sight," which enabies the near cross-hair to be distinctly seen 
at the same time as the distant object. 



Fig. 19. 




(26.) The Telescope Level. In this the line of collimation 
of the telescope corresponds to the sights of Fig. 17, and is 
made parallel to the level ; i. e., this line is so adjusted as to 
be horizontal when the bubble of its level is in the centre. 

There are many different forms of the Telescope Level, of 
which the most important ones will now be given. 

Note. — The level, represented in Fig. 19, and described in the following arti- 
cles, and the Transit, represented in Fig. 73, and described in Art. (92), are made 
by W. & L. E. Gurley, of Troy, K Y., to whom the editor is indebted for valuable 
information respecting the construction and adjustments of the instruments. 



16 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



(27.) The Y Level. This is so named from 
the shape of the supports of the telescope. It 
is the variety most used by American engi- 
neers. 

Fig. 19 represents a twenty-inch Y level 
of the usual form. The telescope is held in 
the wyes by the clips, A A, which are fast- 
ened to the wyes by tapering pins, so that 
the telescope can be clamped in any position. 
The milled-headed screws at M and M are 
used to move the object-glass and eye-piece in 
and out, so as to adjust them for long and 
short sights, and for short-sighted and long- 
sighted persons. L is a spirit-level ; P and P 
are parallel plates; C is the clamp-screw, 
which fastens the spindle on which the level 
bar, B, which supports the wyes, turns ; T is 
the tangent screw, by which the telescope may 
be slowly turned around horizontally. 



Fig. 




BC w r ~w 



C J 



(28.) The Telescope. The arrangement of 
the parts of the telescope is shown in Fig. 20. 
O is the object-glass, by which an image of 
any object, toward which the telescope may 
be directed, is formed within the tube. E E is 
the eye-piece — a combination of lenses, so ar- 
ranged as to magnify the small image formed 
by the object-glass. The cross-hairs are at X. 
They are moved by means of the screws shown 
at B B. A A are screws used for centring the 
eye-piece. C C are screws used for centring 
the object-glass. At D D are rings, or collars, 
of exactly the same diameter, turned very truly, 
by which the telescope revolves in the wyes. 

The telescope shown in the figure forms the image erect. 
Other combinations of lenses are used, some of which invert 
the image ; but the one here shown is generally preferred. 



i c 



u 



isp 



AIR-BUBBLE ,0R SPIRIT LEVELS. 



17 



Fig. 21. 




(29.) The Cross-hairs. These are made of very fine pla- 
tinum wire or of spider-threads. They are attached to a 

short, thick tube, placed 
within the telescope- 
tube, through which pass 
loosely four screws whose 
threads enter and take 
hold of the cross-hair 
ring, as shown in Fig. 
21. 

In some instruments, 
one of each pair of op- 
posite screws is replaced 
by a spring; and the 
screws, instead of being capstan-headed, and -moved by an 
" adjusting -pin," have square heads, and are moved by a 
" key," like a watch-key. 

The line of collimation (or line of aim) is the imaginary 
line passing through the intersection of the cross-hairs and the 
optical centre of the object-glass. 

The image formed by the object-glass should coincide pre- 
cisely with the cross-hairs. When this is not the case, there 
will be an apparent movement of the cross-hairs, about the 
objects sighted to, on moving the eye of the observer. This 
is called instrumental parallax. To correct it, move the eye- 
piece out or in, till the cross-hairs are sharply defined against 
any white object. Then move the object-glass in or out, till 
the object is also distinctly seen. The image is now formed 
where the cross-hairs are, and no movement of the eye will 
cause any apparent motion of the cross-hairs. 



(30.) The Level. This consists of a thick glass tube, slightly 
curved upward, and so nearly filled with alcohol that only a 
small bubble of air remains in the tube. This always rises to 
the highest part. The brass case, in which this is enclosed, is 
attached to the under side of the telescope, and is furnished 
with the means of moving, at one end vertically, and at the 
2 



18 LEVELLING, TOPOGRAPHY, AN£ HIGHER SURVEYING. 

other, horizontally. Over the aperture, in the case, through 
which the bubble-phial is seen, is a graduated level-scale, num- , 
bered each way from zero at the centre. 

(31.) Supports. The wyes in which the telescope rests, are 
supported by the level-bar, B, and fastened to it by two nuts 
at each end (one above and one below the bar), which may be 
moved with an adjusting-pin. The use of these nuts will be 
explained under " Adjustments." Attached to the centre of 
the level-bar is a steel spindle, made so as to turn smoothly 
and firmly in a hollow cylinder of bell-metal ; this, again, is 
fitted to the main socket of the upper parallel plate. 

(32.) Parallel Plates. It is by the aid of these that the 
instrument is levelled. The plates are united by a ball-and- 
socket joint, and are held apart by the four plate-screws, 
Q Q Q Q, which pass through the upper one, and press against 
the lower one. 

To level the instrument, turn the telescope till it is brought 
over a pair of opposite parallel plate-screws. Then turn the 
pair of screws, to which the telescope has been made parallel, 
equally in opposite directions, screwing one in and the other 
out, till the bubble is brought to the centre. Then turn the 
telescope so as to bring it over the other pair of opposite 
screws, and bring the bubble to the centre, as before. 

Repeat the operation, as moving one' pair of screws may 
affect the other. 

Sometimes one of each pair of opposite screws is replaced 
by a strong spring, and in some instruments only three screws 
are used. 

The lower plate is screwed on to the tripod-head. For 
tripods, see Article (45). 

(33.) Adjustments. The line of collimation of the telescope 
should be horizontal when the bubble is in the centre of the 
tube ; which will be the case when this line is parallel to the 
plane of the level. But both this line and this plane are 



AIR-BUBBLE OR SPIRIT LEVELS. 19 

imaginary, and cannot be compared together directly. They 
are therefore compared indirectly. The line of collimation is 
made parallel to the bottom of the collars, and the plane of 
the level is then made parallel to them. 

(34.) First Adjustment. To make the line of collimation 
parallel to the bottoms of the collars. 

Sight to some well-defined point, as far off as it can be dis- 
tinctly seen. Then revolve the telescope half around in its sup- 
ports ; i. e., turn it upside down. If the line of collimation was 
not in the imaginary axis of the rings, or collars, on which the 
telescope rests, it will now no longer bisect the object sighted to. 
Thus, if the horizontal hair was too high, as in Fig. 22, this line 

Fig. 22. 



of collimation would point at first to A, and, after being turned 
over, it would point to B. The error is doubled by the rever- 
sion, and it should point to C, midway between A and B. 
Make it do so, by unscrewing the upper capstan-headed screw, 
and screwing in the lower one, till the horizontal hair is 
brought half-way back to the point. Bemember that, in an 
erecting telescope, the cross-hairs are reversed, and vice versa. 
Bring it the rest of the way by means of the parallel plate- 
screws. Then revolve it in the wyes back to its original po- 
sition, and see if the intersection of the cross-hairs now bisects 
the point, as it should. If not, again revolve, and repeat the 
operation till it is perfected. If the vertical hair passes to 
the right or to the left of the point when the telescope is 
turned half around, it must be adjusted in the same manner 
by the other pair of cross-hair screws. One of these adjust- 
ments may disturb the other, and they should be repeated 
alternately. When they are perfected, the intersection of the 
cross-hairs, when once fixed on a point, will not move from it 



20 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

when the telescope is revolved in its supports. This double 
operation is called adjusting the line of collimation. 

It has now been brought into the centre line, or axis, of the 
collars, and is therefore parallel to their bottoms, or the points 
on which they rest, if they are of equal diameters. "We have 
to assume this as having been effected by the maker. 

In making this adjustment, the level should be clamped, 
but need not be levelled. 



(35.) Second Adjustment. To make the bottoms of the col- 
lars parallel to the plane of the level ; i. e., to insure their 
being horizontal when the bubble is in the centre. 

Clamp the instrument, and bring the bubble to the centre 
by the parallel plate-screws. Take the telescope out of the 
wyes, and turn it end for end. If the bubble returns to the 
centre, " all right." If not, rectify it, by bringing the bubble 
half-way back, by means of the nuts which are above and 
below one end of the bubble-tube, and which work on a screw. 
Bring it the rest of the way by the plate-screws, and again 
turn end for end. Repeat the operation, if necessary. 

If, in revolving the telescope (as in the first adjustment), 
the bnbble runs toward either end, it must be adjusted side- 
ways, by means of two screws which press horizontally against 
the other end of the bubble-tube. This part of the adjustment 
may derange the preceding part, which must, therefore, be 
tried ag^in. 

(36.) Third Adjustment. To cause the bubble to remain in 
the centre of the tube when the telescope is turned around hor- 
izontally. 

To verify this, bring the bubble to the centre of the tube, 
and then turn the telescope half-way around horizontally. If 
the bubble does not remain in the centre, adjust it by bringing 
it half-way back by means of the nuts at the end of the level- 
bar. Test it by bringing it the rest of the way back by the 
parallel plate-screws, and again turning half-way around. 

The cause of the difficulty is, that the plane of the level is 



AIR-BUBBLE OR SPIRIT LEVELS. 21 

not perpendicular to the axis about which it turns, and that 
this axis is not vertical. The above operations correct both 
these faults. 

This adjustment is mainly for convenience, and not for 
accuracy, except in a very small degree. 

Some instruments have no means of making the third ad- 
justment. They must be treated thus : 

Use the screws at the end of the bubble-tube, to cause the 
bubble to remain in the centre when the level is turned around 
horizontally. Then make the line of collimation parallel to 
the level by the method given in Art. (38), by raising or low- 
ering the cross-hairs. 

(37.) The operations of centring the eye-piece and object- 
glass should precede the first three which we have just ex- 
plained. 

Centring the Object-glass. After adjusting the line of 
collimation for a distant object (as explained in the " First 
Adjustment," Art. (34), move out the slide, which carries the 
object-glass, until a point ten or fifteen feet distant can be 
distinctly seen. Then turn the telescope half over, as before, 
and see if the intersection of the cross-hairs bisects the point. 
If not, bring it half-way back by the screws C C, Fig. 20, 
moving only one pair of screws at a time. Repeat the opera- 
tion for a distant point, and then again for a near one, if 
necessary. We have now adjusted the line of collimation for 
long and short sights, and may assume it to be in adjustment 
for intermediate ones, since the bearings of the slides are sup- 
posed to be true, and their planes parallel to each other. 

Centring the Eye-piece. This is to enable the observer 
to see the intersection of the cross-hairs precisely in the centre 
of the field of view of the eye-piece. It is adjusted by means 
of four screws, two of which are shown at A, A. 

These operations are performed by the maker so perma- 
nently as to need no further attention from the engineer, and the 
heads of the. screws, by which these adjustments are made, are 
covered by a thin ring which protects them from disturbance. 



22 levelling; topography, and higher surveying. 

(38.) Adjustment by setting between two points, or the 
" Peg Method." Drive two pegs several hundred feet apart, 
and set the instrument midway between them. Level, and 
sight to the rod held on each peg. The difference of the read- 
ings will be the true difference of the heights of the pegs ; no 
matter how much the level may be out of adjustment. 

Then set the level over one peg, and sight to the rod at 
the other. Measure the height of the cross-hairs above the 
first peg. The difference of this and the reading on the rod 
should equal the difference of the heights of the two points, as 
previously determined. If it does not, set the target to the 
sum or difference of the height of the cross-hairs above the 
first peg, and the true difference of height of the points, ac- 
cording as the first point is higher or lower than the second, 
and hold the rod on the second point. Sight to it, and raise 
or lower one end of the bubble-tube until the horizontal cross- 
hair does bisect the target when the bubble is in the centre. 
Then perform the " third adjustment." 

Instead of setting over one peg, it is generally more con- 
venient to set near to it, and sight to a rod held on it, and use 
this reading, instead of the measured height of the cross-hairs. 

Fig. 23. 
2.301* 
1.7051- 



,-.693 

!N". B. This verification should always be used for every 
level, even after the three usual adjustments have been made ; 
for it is independent of the equality of the collar^. 

In running a long line of levels, let the last sight at night 
be taken midway between the last two " turning-point " 
pegs, and in the morning try their difference by setting close 
to the last one. This tests the level every day with very little 
extra labor. 



AIR-BUBBLE OR SPIRIT LEVELS. 



23 



(39.) Verification oy another Telescope. Set up and level 
the instrument, and bisect the target on a distant rod. Then 
turn the telescope half around horizontally, and bring the 
bubble to the centre, if disturbed. Then take another tele- 
scope, of about the same magnifying power, but with a larger 
object-glass. Hold it close to the object-glass of. the level, and 
look through the level telescope. You will see the cross-hairs 
plainly, and by the side of your telescope you will see the 
target. If the cross-hairs bisect the target, " all right." If 
not, adjust as in last method. If the second telescope be not 
larger than that of the level, hold it to one side. 

(40.) Egault's Level. In this level the bubble-tube is not 
connected with the telescope. It is used thus : 

FlG ^ Level, and sight as 

usual. Then turn the 
telescope upside down, 
end for end, and half 
way around horizon- 
tally, and sight again. 
Half the sum of the 
two readings is the 
correct one, no matter 
how much the instrument is out of adjustment (assuming the 
collars to be of equal size) ; for the errors then cancel each 
other. This is the one used principally in France. 

The rod used with it is marked with numbers only half the 
real heights above its bottom. Then the sum of the readings 
is the true one. Thus the rod itself takes the mean of the 
readings. 

Fig. 25. 





(41.) Troughton's Level. In this the bubble-tube is perma- 
nently fastened in the top of the telescope-tube. It is adjusted 



24 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



by the "peg method," or some similar one, the cross-hair 
being moved up or down until the observation gives the true 
difference of height of the pegs when the bubble is in the 
centre. Then make the " Third Adjustment," by means of 
the screws under the telescope. 



Fig. 26. 



(42.) Gravatt's Level, or the "Dumpy Level, 
is very great, thus giv- 
ing more light. Its 
bubble is on the top, 
and can be seen in a 
small inclined mirror, 
by the observer. It al- 
so has a cross-level. 



Its diameter 




(43.) Bourdaloue's 
Level. This is a modifi- 
cation of Egault's. The 
telescope carries a steel 
prism near each end ; 
one of which rests on a knife-edge, and the other on the 
spherical top of an adjusting-screw. 

(44.) Lenoir's Level. In this, the telescope carries, at each 
end, a steel block, whose upper and lower faces are made very 

Fig. 27. 




perfectly parallel. They are placed on a brass circle, which 
is made level by reversing a level placed upon the telescope. 



RODS. 



25 



(45.) 



Fig. 



Fig. 29. 



Tripods. These consist of three legs, shod with iron, 
and connected by joints at the 
top. There are many different 
forms, the most common of which 
is given in Fig. 19. Other forms 
are given in Figs. 26, 28, and 29. 
Lightness and stiffness are the de- 
sired qualities. Of the two rep- 
resented in Figs. 28 and 29, the 
first has the advantage of being 
simple and cheap ; and the second 
of being light and yet strong. 

Stephenson's tripod has a ball- 
and-socket joint below the par- 
allel plates, so as to admit of 
being at once set nearly level on 
very steep slopes. 




CHAPTER Y. 



(46. 

Fig. 33 



Fig. 31 

a 



EODS. 

) These should be made of light, well-seasoned wood. 
A plumb or level attached to them will 
show when they are held vertically. To 
detect whether the rod leans to or from the 
instrument, its front may be angular or 
curved. If angular, when held leaning tow- 
ard the instrument, the lines of division 
will appear as in Fig. 30. When leaning 
from the instrument, they will appear as in 
Fig. 31. They are usually divided to feet, 
tenths, and hundredths. 



26 



LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



(47.) Target. This is a plate of iron or brass, attached to 
the rod in such a way that it may be moved up and down the 
rod and clamped in any position. The face of the target 
should be painted of such a pattern that, when sighting to it, 
it may be very precisely bisected by the horizontal cross-hair. 
Some of the many varieties are given in Figs. 32-40. 



Fig. 32. 



Fig. 33. 



Fig. 34. 



w-W 



Fig. 35. 



Fig. 36. 



Fig. 37. 





® 



Fig. 38. 



Fig. 



Fig. 40. 





o 




Those represented in Figs. 32, 33, and 34, are bad, because 
the cross-hair may be above or below the middle of the target 
by its full thickness, as magnified by the eye-piece of the tel- 
escope, without the error being perceptible. The next three, 
Figs. 35, 36, and 37, depend upon the nicety with which the 
eye can determine if a line bisects an angle. Fig. 38 depends 
upon the accuracy with which the eye can bisect a space. 
Fig. 39 depends upon the accuracy with which the eye can 
bisect a circle. Figs. 36, 37, and 40, are the best forms for 
use. Red and white are the best colors. 

A good method of moving the target on a long rod, is by 
means of pulleys at the ends of the rod. A woollen cord 



KODS. 



27 



Fig. 41. 




should be used, on account of its being least 
affected by moisture. 

(48.) Vernier's. L. S. [343-357]. The target 
carries a Vernier, by which smaller spaces may 
be measured than those into which the rod 
is divided. It may be placed on the side of an 
aperture, in the face of the target, through 
which the divisions on the rod can be seen ; or 
carried on the back or side of the rod by the 
target-clamp. 



(49.) The New-York Rod. This is in two 
pieces, sliding one upon the other, and con- 
nected together by a tongue. It is graduated 
to tenths and hundredths of a foot, and can be 
read to thousandths by the Vernier. Up to six 
feet the target is used as on other rods. For 
greater heights, the target is fixed at 6 \ ft., and 
the back part of the rod, which carries the tar- 
get, is shoved up (Fig. 41) until the target is 
bisected by the cross-hairs. Its height is then 
read off on the side of the rod ; on which the 
numbers run downward, and on which is a 
second vernier, which gives the 
precise reading. It is convenient 
for its portability, but apt to bind 
or slip in sliding ; i. e., to be too 
tight or too loose, as the weather 
is moist or dry. 



Fig. 43. 



w 



(50.) The Boston Rod. This is m 
two parts like the New York rod. 
The target is rectangular (Fig. 42), 
and is fastened to one of the pieces near its extremity. For 
heights less than six feet, the rod is held with the target-end 
down, and the target is moved up by sliding up the piece 



28 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

which carries it. For heights above six feet, the rod is turned 
end for end, bringing the target-end up, and then sliding up 
the piece which carries the target. 

(51.) Speaking-Rods. These are rods which are read with- 
out targets, the divisions and subdivisions being painted on 



Fig. 45. 



Fig. 43. 



Fio. 44. 



mummmmmaaa 
« 



the face of the rod. They produce great saving of time and 
increase of accuracy. 

In one form, Fig. 43, the face of the rod is divided into 
tenths of feet, and smaller divisions estimated. 

In Bourdaloue's rod the divisions are each 4 centimetres 
(1.6 inches), and are numbered at half their value. He 
arranges them as in Fig. 44. 

Grmatt's Rod, Fig. 45. This is divided to 0.01 foot. The 
upper hundredth of each tenth extends across the rod. Each 
half-tenth is marked by a dot. Each half-foot by two dots. 
Every other tenth is numbered, and the numbers are each 



HODS. 



29 



0.1 high. It is in three parts, which slide into each other like 
a telescope. 

Barlow's Bod, Fig. 46. In this the divisions are marked 
by triangles, each 0.02 ft. high, so that it reads to hundredths, 
and less by estimation. This is based on the power the eye 
has in bisecting angles. 

Stephenson's Bod, Fig. 47. This is based upon the prin- 
ciple of the Diagonal Scale. Each tenth is bisected by a hor- 
izontal line, and the diagonals enable the observer to read to 
hundredths. ■ 

Conybewre's Bod, Fig. 48. It reads to hundredths of a 
foot by means of the cross-hair bisecting the tops and bottoms 
and angles of hexagons. The odd tenths are made white and 
the even ones black. The figures are placed so that their 
centres are opposite the divisions they refer to. 

Pembertorfs Bod, Fig. 49. This is on the principle of 9 

Fig. 47. 



Fig. 49. 





Fig. 46. 


► 


► 




verniers placed side by si'de. It reads to hundredths, which 
are given by counting up from the dot which the hair bisects, 
to the dot in the same vertical line which is bisected by one 



30 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

of the horizontal lines which mark the tenths. The inventor 
claims that it can be read 9 times as far as Gravatt's. 

On all speaking-rods, to avoid confounding numbers, such 
as 3 and 8, they may be marked thus : 

1 . 2 . Ill . 4 . Y . 6 . 7 . 8 . IX . X . 11 . XII. 

The French, who go by tenths, use the following : 

I.2.T.4.V.6.7.8. N . X. 

The figures are sometimes placed with their tops on a 
level with the tops of the dimensions they mark, e. g., feet ; 
and sometimes with their middles on the dividing line. 



CHAPTEK VI. 



THE PRACTICE 



(52.) Field Routine: or, how to start and go on. 

1. The rodman holds the rod on the starting-point ; which 
may be a peg, a door-sill, or other " bench-mark " Art. (60). 
He stands square behind his rod, and holds it as nearly ver- 
tical as possible. . 

2. The leveller sets up the instrument, somewhere in 
the direction in which he is going, but not necessarily, or 
usually, in the precise line. He then levels the instrument 
by the parallel plate-screws, sights to the rod, and notes the 
reading, whether of target or speaking-rod, as a " back-sight " 
(B. S.), or + (plus) sight ; entering it in the proper column 
of one of the tabular forms of field-book, given in the follow- 
ing articles. 

3. The rodman is then sent ahead about as far as he was 
behind, and he there drives a "level-peg" nearly to the sur- 



THE PRACTICE. 31 

face of the ground, or finds a hard, well-defined point, and 
holds the rod upon it. 

4. The leveller then again sights to the rod, and notes 
the reading as a " fore-sight " (F. S.), or — (minus) sight. The 
difference of the two readings is the difference of the heights 
of the points. 

5. He then takes up the instrument, goes beyond the rod, 
any convenient distance, sets up again, and proceeds as in 
paragraph 2 ; and so on for any number of points, which will 
form a series of pairs. The successive observations of each 
pair give their difference of heights, and the combination of 
all these gives the difference of heights of the first and last 
points of the series. 

6. If the vertical cross-hair be strictly vertical, it will de- 
termine whether the rod leans to the right or left. To know 
whether the cross-hair is vertical or not, try whether it coin- 
cides with a plumb-line ; or sight to some fixed point, turn 
the telescope from side to side horizontally, and see if the hor- 
izontal cross-hair continues to cover the spot. If it does not, 
turn the telescope around in the wyes till it does ; then it is 
truly horizontal, and the other hair, being perpendicular to it, 
is truly vertical. To know whether the rod leans forward or 
backward, have the rodman move it from and to himself. If 
the line bisected by the cross-hair descends in both motions, 
the rod was vertical. If the line rises-, the rod was leaning. 
The lowest reading is the true one. 

7. When a target is used, signals are made by the leveller 
with the hand, " up " and " down," to indicate in which 
direction to move the target. Drawing the hand to the side 
signifies " stop," and both hands brought together above the 
head signifies " all right." The rodman should move the tar- 
get fast at first, and slowly after having passed the right point. 
When signalled " all right," he should clamp the target and 
show again. Then call out the reading before moving, and 
show it to the leveller, as either passes the other. 

8. We have thus far supposed that only the difference of 
heights of the two extreme points is desired. But when a 



32 LEVELLING, TOPOGEAPHY, AND HIGHER SURVEYING. 

section or profile of the ground is required, the rod must be 
held and observed, at each change of slope of the ground ; or 
at regular distances ; usually, for railroad work, at every hun- 
dred feet, and also at any change of slope between those 
points. 

Any number of points, within sight, may have their relative 
heights determined at one setting of the level. 

The names back-sight (B. S.) and fore-sight (F. S.) do not 
necessarily mean sights taken looking forward or backward 
(though they are generally so for turning-points), but the first 
sight taken, after setting up the instrument, is a B. S. or + 
(plus) sight, and all following ones, taken before removing 
the instrument, are F. S.'s, or — (minus) sights. . The full 
meaning of this will appear in considering the forms of field* 
book. 

All but the first and last points sighted to are called mter* 
mediate points, or " intermediates." The last point sighted 
to before moving the instrument is called a turning-point or 
changing-point. 

The first and last sights, taken at any one setting of the 
instrument, require the greatest possible accuracy. The in- 
termediate points may be taken only to the nearest tenth, or 
hundredth at most ; because any error in them will not affect 
the final result, but only the height of that single point at 
which it was taken. 

Two rodmen are often used to save the time of the leveller. 
Then it is well to use a target-rod for the " turning-points," 
which are often distant and need most precision, and a speak- 
ing-rod for the intermediate points. Where one rod is used, 
the rodman should keep notes of the readings at the turning- 
points. 

(53.) Field-notes. The beginner may sketch the heights 
and distances measured, in a profile or side view,, as in Fig. 
50. But when the observations are numerous, they should be 
placed in one of the tabular forms given on the following 
pages. 



THE PRACTICE. 

Fig. 50. 



33 




(54.) First Form of Field-book. In this, the names of the 
points, or " Stations," whose heights are demanded, are placed 
in the first column ; and their heights, as finally ascertained, in 
reference to the first point, in the last column. The heights 
above the starting-point are marked + , and those below it are 
marked — . The back-sight to any station is placed on the 
line below the point to which it refers. When a back-sight 
exceeds a fore-sight, their difference is placed in the column 
of " Eise ; " when it is less, their difference is a " Fall." The 
following table represents the same observations as the last 
figure, and their careful comparison will explain any obscuri- 
ties in either : 



Stations. 


Distances. 


Back-sights. 


Fore-sights. 


Eise. 


Fall. 


To. Heights. 


A 












0.00 


B 


100 


2.00 


6.00 




— 4.00 


— 4.00 


C 


60 


3.00 


4.00 




— 1.00 


— 5.00 


D 


40 


2.00 


1.00 


4- 1.00 




— 4.00 


E 


70 


6.00 


1.00 


+ 5.00 




+ 1.00 


F 


50 


2.00 


6.00 




-4.00 


— 3.00 


15.00 


18.00 


— 3.00 



The above table shows that B is 4 feet below A ; that C is 
5 feet below A ; that E is 1 foot above A ; and so on. To test 
the calculations, add up the back-sights and fore-sights. The 
difference of the sums should equal the last " total height." 
3 



34: LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



An objection to this form is that the back-sights come on 
the line below the station to which they are taken, which is 
embarrassing to a beginner. 

When " intermediate " observations are taken, the " fore- 
sights," taken to these intermediate points, are pnt down. in 
their proper column, and are also set down in the column of 
" back-sights ; " so that when the two columns are added up, 
any error in these intermediate sights (which are usually not 
taken very accurately) will be cancelled, and will not affect 
the final result. The effect is the same as if, after the fore- 
sight to the intermediate point had been taken, the instrument 
had been taken up and set down again at precisely the same 
height as before, and a back-sight had then been taken to the 
same point. Hence, in this form, the " turning-points " are 
those stations which have different back-sights and fore-sights, 
while those which have them the same are " intermediates." 

The following figure and table represent the same ground 
as the preceding one, but with only two settings of the instru- 
ment. D is the turning-point : 

Fig. 51. 




Stations. 


Distances. 


B. S. + 


F. S. - 


Eise. 


Fall. 


To. Heights. 


A 












0.00 


B 




2.00 


6.00 




4.00 


— 4.00 


C 




6.00 


7.00 




1.00 


— 5.00 


D 




7.00 


6.00 


1.00 




— 4.00 


E 




9.00 


4.00 


5.00 




+ 1.00 


P 




4.00 


8.00 




4.00 


— 3.00 


+ 28.00 


— 31.00 


3.00 



THE PRACTICE. 



35 



In levelling for "sections," the distances between the 
points levelled mnst be recorded. They are nsnally put down 
after the stations to which they are measured ; although in 
surveying with the compass, etc., they are put down after the 
stations from which they are measured. In the following 
notes, which contain intermediate stations, they are put down 
oefore the stations to which they are measured. It should be 
remembered that these distances are measured between the 
points at which the rod is held, and have no reference to the. 
points at which the instrument is set up : 



Distance. 


Stations. 


B. S. + 


F. S. - 


Eise. 


Fall. 


To. Heights. 




260 










91.397 


100 


261 


4.576 


3.726 


0.850 




92.247 


100 


• 262 


5.420 


4.500 


0.920 




93.167 


100 


263 


4.500 


3.170 


1.330 




94.497 


40 


263.40 


4.910 


4.938 




■ 0.028 


94.469 


60 


264 


4.938 


6.386 




1.448 


93.021 


100 


265 


3.380 


4.640 




1.260 


91.761 


100 


266 


* 4.640 


5.400 




0.760 


91.001 


70 


266.70 


2.760 


3.070 




0.310 


90.691 


30 


267 


3.070 


3.750 




0.680 


90.011 


100 


268 


3.750 


6.925 




3.175 


86.836 , 


41.944 


46.505 


-4.561 








41.944 




+ 91.397 




- 4.561 


86.836 



(55.) Second Form of Field-book. This is presented below. 
It refers to the same stations and levels noted in the first 
table, and shown in Fig. 50 : 



Stations. 


Distances. 


Back-sights. 


Ht. Inst, above Datum. 


Fore-sights. 


To. Heights; 


A 










0.00 


B 


100 


2.00 


+ 2.00 


6.00 


- 4.00 


C 


60 


• 3.00 


— 1.00 


4.00 


— 5.00 


D 


40 


2.00 


— 3.00 


1.00 


— 4.00 


E 


70 


6.00 


+ 2.00 


1.00 


+ 1.00 


F 


50 


2.00 


+ 3.00 


6.00 


- 3.00 


15.00 


18.00 


— 3.00 



36 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

In the preceding form it will be seen that a new column is 
introduced, containing the Height of the Instrument (i. e., of 
its line of sight), not above the ground where it stands, but 
above the Datum, or starting-point, of the levels. The former 
columns of " Kise " and " Fall " are omitted. The preceding 
notes are taken thus : The height of the starting-point, or 
" datum," at A, is 0.00. The instrument being set up and 
levelled, the rod is held at A. The back-sight upon it is 2.00 ; 
therefore the height of the instrument is also 2.00. The rod 
is next held at B. The fore-sight to it is 6.00. That point is 
therefore 6.00 below the instrument, or 2.00 — 6.00 = — 4.00 
below the datum. The instrument is now moved, and again 
set up, and the back-sight to B, being 3.00, the height of the 
instrument is — 4.00 + 3.00 — — 1.00, and so on ; the height 
of the instrument being always obtained by adding the back- 
sight to the height of the peg on which the rod is held, and 
the height of the next peg being obtained by subtracting the 
fore-sight to the rod held on that peg, from the height of the 
instrument. 

This form is better than the first form, in levelling for a 
section of the ground to make a profile ; or when several ob- 
servations are to be made at one setting of the level ; or when 
points of desired heights are to be established, as in " Level- 
ling-location," Chapter Till. 

This form may be modified by putting the back-sights on 
the same line with the stations to which they are taken. This 
avoids the defect of the first form, but introduces the new 
defect of writing them down after the number which they 
precede, in a back-handed way, which may be a source of 
error. 

This modification is shown in the following table, which 
corresponds to Fig. 51. In the column of fore-sights, the 
" turning-points " (T. P.), and " intermediate points " (Int.), 
are put in separate columns ; so that, to prove the work, the 
difference of the sum of the back-sights and of the sum of the 
turning-point fore-sights, is the number which should equal 
the difference of the heights of the first and last points : 



THE PRACTICE. 



37 



Stations. 


Distances. 


B. S. + 


Ht. of Inst. 


F. S. - 


To. Heights. 


T. P. 


Int. 


A 
B 

C 
D 
E 
F 




2.00 
9.00 


+ 2.00 
+ 5.00 


6.00 
8.00 


6.00 
7.00 

4.00 


0.00 

— 4.00 

— 5.00 

— 4.00 

+ i.oo 

— 3.00 


+ 11.00 


- 14.00 
+ 11.00 


— 3.00 



When a line is divided up into stations of 100 feet each, 
as on railroad work, the number of the station indicates its 
distance from the starting-point. When an observation is 
taken at a point between these hundred-feet stations, it is noted 
as a decimal thus: Station 4.60 is 460 feet from the starting- 
point. In the field-notes of such work, the column of distances 
may be omitted, as in the following table. The heights and 
distances are the same as in the last table under Art. (54) : 









F. g 






Stations. 


B. S. + 


Ht. of Inst. 


T. P. 


Int. 


Total Heights. 


260 


4.576 


95.973 






91.397 


261 


5.420 


97.667 


3.726 




92.247 


262 








4.500 


93.167 


263 


4.910 


99.407 


3.170 




94.497 


263.40 








4.938 


94.469 


264 


3.380 


96.401 


6.386 




93.021 


265 








4.640 


91.761 


266 


2.760 


93.761 


5.400 




91.001 


266.70 








3.070 


90.691 


267 








3.750 


90.011 


268 






6.925 




86.836 


+ 21.046 


— 25.607 








+ 21.046 






— 4.561 








+ 91.397 






+ 86.836 



38 



LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



(56.) Third Form of Field-book. In this form, the defects 
of the preceding methods are avoided, and it approximates to 
a sketch of the operations ; the fore-sights being placed before 
the stations to which they are taken, and the back-sights after 
them. The distances are placed before the stations to which 
they are taken ; or after those from which they are taken. 
Another advantage is that the stations, their heights, and the 
distances, are brought together ; which facilitates the making 
of a profile. The following table is the case given in Fig. 50 : 






F. S. - 


Distances. 


Stations. 


Ht. of Peg. 


B. S. + 


Ht. of Inst. 






A 


0.00 


2.00 


+ 2.00 


6.00 


100 


B 


— 4.00 


3.00 


— 1.00 


4.00 


60 


C 


— 5.00 


2.00 


— 3.00 


1.00 


40 


D 


— 4.00 


6.00 


+ 2.00 


1.00 


10 


E 


+ 1.00 


2.00 


+ 3.00 


6.00 


50 


F 


— 3.00 






— 18.00 


+ 15.00 
— 18.00 


- 3.00 



When " intermediates " are taken, the first column may be 
divided into two heads (as in the second table, Art. 55), re- 
spectively "turning-points" (T.P.), and " intermediate points" 
(Int.). The work is tested by taking the difference of the sum 
of the " T. P.'s " and " B. S.'s " The symbol © is used to rep- 
resent the height of the cross-hairs. This table is for Fig. 51 : 



F. £ 










B. S. + 








Stations. 


Distances. 


Ht. of Peg. 


© 


T. P. 


Int. 
















A 


100 


0.00 


2.00 


+ 2.00 




6.00 


B 


60 


— 4.00 








7.00 


C 


40 


— 5.00 






6.00 




1 D 


TO 


— 4.00 


9.00 


+ 5.00 




4.00 


E 


50 


+ 1.00 






8.00 




F 




-3.00 






- 14.00 


+ 11.00 












- 14.00 




— 3.00 



THE PRACTICE. 



39 



Fourth Form of Field-book. In this the back sights are 
placed directly under the height of the station to which they 
are taken, which lessens the chance of making mistakes in 
adding to get the height of instrument. The height of in- 
strument is distinguished by being included between two 
horizontal lines. The following table refers to Fig. 51. 



Station. * 


F. S. 


Heights. 


Eemarks. 


A 

B 
C 
D 

E 
F 


6.00 
7.00 
6.00 

4.00 
8.00 


0.00 
2.00 




2.00 


-4.00 

-5.00 

-4.00 

9.00 


5.00 


1.00 
—3.00 



(57.) Best Length of Sights. There are two classes of inac- 
curacies. With very long sights, the errors of imperfect ad- 
justment and curvature are greatest ; the former varying as 
the length, and the latter as the square of the length. With 
very short sights, and therefore more numerous, the errors of 
inaccurate, sighting at the target are greatest. The best usual 
mean is from 200 feet to 300 feet, or more if equal distances 
for back-sights and fore-sights to turning-points can be ob- 
tained. 

(58.) Equal Distances of Sight. They are always very de- 
sirable. They are most easily determined, when no stakes 
have been previously set, by " stadia " cross-hairs in the tele- 
scope of the level. [L. S., 375.] 

(59.) Patum-Level. This is the plane of reference, from 
which, above it or below it, usually the former, the heights of 
all points of the line are reckoned. 



40 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

It may be taken as the height of the starting-point. If 
the line descends, it is better to call the starting-point 10 feet 
or 100 feet above some imaginary plane, so that points below 
the starting-point may not have minus signs. 

It is desirable to refer all levels in a country to some one 
datum. This is usually the surface of the sea, and for general 
purposes mean tide is best. Low-water mark should be the 
datum when the levellings are connected with harbor-surveys, 
Whose soundings always refer to low water. High-water 
mark should be used when the levellings relate to the drain- 
age of a country. 

(60.) Bench-Marks (B. M.). These are permanent objects, 
natural or artificial, whose heights above the datum are de- 
termined and recorded for future reference. 

Good objects are these : pointed tops of rocks ; tops of 
milestones ; stone door-sills ; tops of gate-posts or hinges ; and 
generally any object not easily disturbed, and easily described 
and found. 

A knob made on the spreading root of a tree is good. A 
nail may be driven in it, and the FlG 52 

tree " blazed" and marked, as in 
Fig. 52. A stake will do till frost. 

Bench-marks should be made 
near the starting-point of a line of 
levels ; near where the line crosses a 
road ; on each side of a river crossed 
by it ; at the top and bottom of any 
high hill passed over ; and always at every half-mile or mile. 

The precise location and description of every B. M. should 
be noted very fully and precisely, and in such a way that an 
entire stranger could find it, with the aid of the notes. 




(61.) Check-Levels, or Test-Levels. No single set of levels 
is to be trusted ; but they must be tested by another set, run 
between the bench-marks (B. M.'s), though not necessarily 
over the same ground. 



THE PRACTICE. ^ 

A set of levels will verify themselves if they come around 
to the starting-point again. 

(62.) Limits of Precision. Errors and inaccuracies should 
be carefully distinguished. For the latter, every leveller 
must make a standard for himself, so as to be able, in testing 
his work, to distinguish any real error from his usual inac- 
curacy. 

The result of four sets of levellings, in France, of from 45 
to 140 miles, averaged a difference of t l ft. in 43 miles, and 
the greatest error was \ ft. in 56 miles. 

A French leveller, M. Bourdaloue, contracts to level the 
B. M.'s of a B. B. survey to within 0.002 ft. per mile, or -^ ft. 
per 50 miles. 

In Scotland, the difference of two sets of levels of 26 miles 
was 0,02 ft. 

(63.) Trial-Levels, or Flying-Levels. Their object is to get 
a general approximate idea of the comparative heights of 
a portion of the country, as a guide in choosing lines to be 
levelled more accurately. More rapidity is required, and less 
precision is necessary. The distances may be measured at the 
same time by stadia-hairs. 

(64.) Levelling for Sections. The object of this is to meas- 
ure all the ascents and descents of the line, and the distances 
between the points at which the slope changes; so that a 
section or profile of it can be made from the observations taken. 

The line of a railroad is usually set out by a party with 
compass or transit, who drive at every hundred feet a large 
stake with the number of the station on it, and beside it a 
small level-peg, even with the surface of the ground. On this 
the rod is held for the observations. The level-peg is set in 
" line," and the large stake a foot or two to one side. 

(65.) Profiles. A profile is a section of ground by a verti- 
cal plane or cylindrical surface, 1 passing through the line along 

1 A cylindrical surface is here understood to mean that formed by a line mov- 
ing parallel to itself along any line, instead of only a circle, as in elementary 
geometry. 



42 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

which a profile is desired. It represents to any desired scale 
the heights and distances of the various points of a line, its 
ascents and descents, as seen in a side view. It is made thus : 
Any point on the paper being assumed for the first station, a 
horizontal line is drawn through it ; the distance to the next 
station is measured along it, to the required scale; at the ter- 
mination of this distance a vertical line is drawn ; and the 
given height of the second station above or below the first is 
set off on this vertical line. The point thus fixed determines 
the second station, and a line joining it to the first station 
represents the slope of the ground between the two. The pro- 
cess is repeated for the next station, etc. 

But the rises and falls of a line are always very small in 
proportion to the distances passed over, even mountains being 
merely as the roughnesses of the rind of an orange. If the dis- 
tances and the heights were represented on a profile to the 
same scale, the latter would be hardly visible. To make them 
more apparent, it is usual to " exaggerate the vertical scale " 
tenfold, or more ; i. e., to make the representation of a foot 
of height ten times as great as that of a foot of length, as in 
Tig. 50, in which one inch represents one hundred feet for 
the distances, and ten feet for the heights. 

In practice, engraved profile-paper is generally used, which 
is ruled in squares or rectangles, to which any arbitrary values 
may be assigned. 

"When the line levelled over is not straight, the profile, 
whose length is that of the line straightened out, will extend 
beyond the " plan " when both are on the same sheet. 

(66.) Cross-Levels. These show the heights of the ground 
on a line at right angles to the main line. They give " cross- 
sections " of it. In the note-book they are put on the right- 
hand page. They may be taken at the same time with the 
other levels, or independently. In taking cross-levels where 
the slopes are quite steep, as in mountain districts, frequent 
settings of the instrument are necessary. 

A much more rapid method is by the use of " cross-sec- 



DIFFICULTIES. 



43 



tion rods." These are two rods, one of which is about ten or 
twelve feet long, provided with a bubble-tube near each end, 
so as to be held level, and graduated to feet, tenths, and hun- 
dredths. The other is simply a graduated rod. The manner 
of using them is shown in Fig. 53. 

Fig. 53. 




Aslope-level is sometimes used. See "Angular Survey- 
ing," Part II. 



CHAPTEK YII. 



DIFFICULTIES. 



(67.) Steep Slopes. In descending or ascending a hill, the 
instrument and the rod should be so placed that the sight 
should strike as near as possible to the bottom of the rod on 
the up-hill side, and the top of the rod on the down-hill side. 

Try this by levelling over two screws, setting the instru- 
ment so that one pair of opposite plate-screws shall point in 
the direction of the line, but do not be too particular ; it is a 
waste of time. 

Doing this produces sights of unequal length. The rod 
being about twice as high as the instrument, the down-hill 
sights will be about double the length of the up-hill ones, as 
shown in Fig. 54. Then set to one side of the line. This is 



44 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Fig. 54. 




necessary on slopes so steep that the rod is too near the level 
to be read. If this be impossible, keep notes of the lengths 
of the sights to the turning-points, backward and forward, 
and as soon as possible take sights unequal in the contrary 
direction till the differences ot lengths balance the former 
ones. "When approaching a long ascent or descent, make 
these compensations in advance. 

In levelling over a line of stakes already set, as on a rail- 
road, at every 100 ft., if the line of sight strikes not quite up 
to one, drive a peg as high as you can see it, and make it; a 
turning-point, noting it " peg " in the field-book. 

In levelling across a hill or hollow, instead of setting the 
instrument on the top of the hill or bottom of the hollow, time 
will be saved by the method represented in Figs. 55 and 56. 

Fig. 55. 




(68.) When the rod is a little too low, raise it alongside of 
a stake, or the body, and put the top of the rod "right;" 
then measure down from the bottom of the rod, and add it to 
its length. 



DIFFICULTIES. 
Fig. 56. 



45 




(69.) When the rod is a little too high, so that the line of 
Bight strikes the peg below the bottom of the rod, measure 
down from the top of the peg, and put down the sight with a 
contrary sign to. what it would have had ; i. e., if a back-sight 
make it minus, and if a fore-sight make it plus. 

(70.) When the rod is too near. When no figure is visi- 
ble, raise the rod slowly till a figure comes in sight. If too 
near to read, and there is no target, use a field-book as target. 
If the instrument is exactly over the peg, measure up to the 
height of the cross-hairs, as given by the side-screws. 

(71.) Water. A. — A pond too wide to he sighted across. 
Drive a peg to the level of the water, on the first side, and 
observe its height, as an F. S. Then drive a peg on the other 
side of the pond, also to the surface of the water. Hold the 
rod on it. Set up the level beyond it, and sight to it as a 
B. S., and put down the observation as if it had been taken 
to the first peg. 

Fig. 57. 




46 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



F. S. 


Sta. 


Ht. 


B. S. 


© 


5.0 


74 

74.89 ) 
81.89 y 


50.00 
48.00 


3.00 
6.00 


I 53.00 
54.00 



There must be no wind in the direction of the line of level. 

B. — For levelling across a running stream. Set the two 
pegs in a line at right angles to the current, although the line 
to be levelled may cross it obliquely. 

If a profile or section of the ground under the water be re- 
quired, find the height of the surface, and measure the depths 
below this at a sufficient number of points, measuring the 
distances also, and put these depths down as fore-sights. 

(72.) A Swamp, or Marsh. This cannot be treated like a 
pond, for the water may seem nearly stagnant while its sur- 
face has considerable slope, its flow being retarded by vegeta- 
tion. If only slightly " shaky," have an observer at each end 
of the level. If more so, push the legs down as far as they 
will go, and let both observers lie down on their sides. If still 
more " shaky," drive three stakes or piles, to support the legs 
of the tripod, and stand the tripod on them. 

A water-level will level itself. Use that for intermediate 
points on the swamp, and test the result by levelling around 
the swamp with the spirit-level. 

(73.) Underwood. If it cannot be cut away, set the instru- 
ment on some eminence, natural or artificial. 

(74.) Board Fence. Eun a knife-blade through one of the 
boards, and hold the rod upon it on each side of the fence, as 
if it were a peg, keeping the blade in the same horizontal 
position while the rod and instrument are taken over. 

(75.) A Wall. First Method. Drive a peg at the bottom of 
the wall, on the first side, and observe on it. Measure the 
height of the wall above the peg, and put this down as a B. S. 
Drive another peg on the other side of the wall ; measure down 



DIFFICULTIES. 



47 



to it from the top of the wall, and put that down as an F. S., 
just as if the level had been set in the air at the height of the 
top of the wall, and this B. S. and F. S. had been really taken. 
Set up the instrument beyond the wall, take a B. S. to this 
peg, and go on as usual. 

Fig. 58. 




F. S. 


Sta. 


Ht. 


B. S. 


© 




50 


74.00 


5.00 


79.00 


3.00 


Peg. 


76.00 


13.00 


89.00 


12.00 


Peg. 


77.00 


2.00 


79.00 


1.00 


51 


78.00 







Second Method. Mark where the line of sight strikes the 
wall ; measure up to the top of the wall, and put this down as 
an F. S., with a plus sign, as in (69), where the line of sight 
struck below the top of the peg. 

On the other side of the wall, sight back to it, and mark 
where the line of sight strikes. Measure to the top of the 
wall, and put this down as a B. S., with a minus sign, and 
then go on as usual. 

(76.) House. First try to find some place for the instru- 
ment from which you can see through, by opening doors or 
windows. Or, find some place in the house where you can 
set the instrument and see both ways, or hold the rod at some 
point inside, and look to it from front and back. A straight 
stick may be used if the rod cannot be held upright, and the 
height measured on the rod. 

(77.) The Sun. It often causes the leveller much difficulty. 
1. By shining in the object-glass. If the instrument has a 



48 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



shade on it, draw it out. If not, shade the glass with your 
hand or hat, or set the instrument to one side of the line. 

2. By heating the level unequally in all its parts. Hold- 
ing an umbrella over it will remedy this. 

3. By causing irregular refraction. Some parts of the 
ground become heated more than others, and therefore rarefy 
the air at those places. This cannot be avoided nor corrected. 

(78.) Wind. Watch for lulls of wind, and observe then sev- 
eral times, and take the mean. The least wind is at daybreak. 

(79.) Idiosyncrasies; Different persons do not see things 
precisely alike. Each individual may have an inaccuracy 
peculiar to himself. One may read an observation higher or 
lower than another equal in skill. Also, a person's right and 
left eye may differ. This difference in individuals is termed 
their "personal equation." 

To test the accuracy of your eye, turn the head so as to 
bring the eyes in the same vertical line, and sight to the rod 
held horizontally. Note where the vertical hair strikes. Then 
turn the head to the other side, so as to invert the position of 
the eyes, and then sight again. As before, the mean of the 
two readings is the correct one. 

(80.) Reciprocal Levelling, This is to be used when it is 
impossible to set midway between the two points. 

Fig. 59. 







Set the instrument over A, and sight to rod at B, and note 
reading. The difference of the reading and of the height of 



LEVELLING LOCATION. 49. 

the cross-hairs gives a difference of height of A and B. Then set 
up at B, and observe to A, similarly. A new difference of 
height is obtained. . The mean of these two is the correct one. 

Ht. of cross-hairs above peg at A = 4'. 3 Ht. of cross-hairs above peg at B = 4'.9 
Observation to B = V'.O Observation to A = 4'. 2 

Diff. of height = 2'.Y Diff. of height = O'.T 

True difference = \ (2'.7 + O'.l) = l'.T. 

Otherwise, set the instrument at an equal distance from 
each point, as A' and B', and observe to each in turn. The, 
mean of the two differences of height obtained will be the true 
difference,, as before. 



CHAPTEK VIII. 

LEVELLING LOCATION. 

(81.) Its Nature, It is the converse of the general problem 
of levelling, which is to find the difference of heights of two 
given points. This consists in determining the place of a 
point of any required height above or below any given point. 

To do this, hold the rod on some point of known height 
above the datum-level; sight to it, and thus determine the 
height of the cross-hairs. Subtract from this the desired 
height of the required point, and set the target at the differ- 
ence. Hold the rod at the place where the height is desired, 
and raise or lower it till the cross-hair bisects the target. Then 
the bottom of the rod is at the desired height. Usually, a peg 
is driven till its top is at the given height above the datum. 

(82.) Difficulties. If the difference of height be too much 
to be measured at one setting of the instrument, take a series 
4 



50 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

of levels up or down to the desired point. So, too, if they be 
far apart ; and thus find a place where, the instrument having 
a known height of cross-hairs, the target can finally be set, as 
before. 

If the ground be so low or so high that a peg cannot be 
set with its top at the required height, drive a peg till its top 
is just above the surface of the ground. Observe to the rod 
on it, determine its height above or below the desired point, 
and note this on a ]arge stake driven beside it ; or, place its 
top a whole number of feet above or below the required height, 
and mark the difference on it, or on a stake beside it. 




lill 12 or N ' 



with the words 



(83.) Staking out Work, When embankments and excava- 
tions are to be made for roads, etc., side-stakes are set at points 
in their intended outside 
edges; i. e., where their 
slopes will meet the sur- 
face of the ground; and 
the height which the 
ground at those points is 
above or below the re- 
quired height or depth of 
the top or bottom of the 
finished work, is marked on these stakes 
" cut," or " fill," or the signs -f- or — . 

The places of the stakes 
are found by trial. (See 
Gillespie's Eoad-making, 
p. 145.) These stakes are 
.set to prepare the work 
for contractors. When the 
work is nearly finished, 
other stakes are set at the 
exact required height. 

In staking out foundation-pits, set temporary stakes ex- 
actly above the intended bottom angles of the completed pit, 
thus marking out on the surface of the ground its intended 



Fig. 61. 




LEVELLING LOCATION. 51 

shape. Take the heights of each of these stakes and move 
them outward such distances that cutting down from them 
with the proper depth and slope will bring you to the desired 
bottom angle. 

(84.) To locate a Level-Line. This consists in determining 
on the surface of the ground a series of points which are at 
the same level ; i. e., at the same height above some datum. 
Set one peg at the desired height, as in (81). Sight to the rod 
held thereon, and make fast the target when bisected. Then 
send on the rod in the desired direction, and have it moved up 
or down along the slope of the ground, until the target is 
again bisected. This gives a second point. So go on as far 
as sights can be correctly taken, keeping unchanged the in- 
strument and target. Make the last point sighted to a " turn- 
ing-point." Carry the instrument beyond it, set up again, 
take a B. S., and proceed as at first. 

The rod should be held and pegs driven at points so near 
together that the level-line between them will be approxi- 
mately straight. 

(85.) Applications. One use of this operation is to mark 
out the line which will be the edge of the water of a pond to 
be formed by a dam. In that case, a point of a height equal 
to that of the top of the proposed dam, phis the height which 
the water will stand on it (to be determined by hydraulic 
formulas), will be the starting-point. Then proceed to set 
stakes as directed in the last article. 

The line from stake to stake may then be surveyed like 
the sides of a field, and the area to be overflowed thus de- 
termined. 

Strictly, the surface of the water behind a dam is not level, 
but is curved concavely upward, and is therefore higher and 
sets back farther than if level. This backing up of the water 
is called Remous. 

Another important application of this problem is to obtain 
" contour lines " for Topography. 



52 



LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



(86.) To run a Grade-Line. This consists in setting a series 
of pegs so that their tops shall be points in a line which shall 
have any required slope, ascending or descending. 

When a grade-line is to be run straight between two given 
points, set the level over one point, set the target at the 
height of the cross-hairs, hold the rod on the other point, and 
raise or lower one end of the instrument till the cross-hair 
bisects the target. Then send the rod along the line, and 
drive pegs to such heights that when the rod is held on them 
the cross-hair will bisect the target. A stake may be driven 
at the extreme point to the height of the target. 

A line of uniform 
grade or slope is not a 
straight line. Calling 
the globe spherical, 
this line, when traced 
in the plane of a great 
circle, would be a log- 
arithmic spiral. On a 
length of six miles, the difference in the middle between it and 
its straight chord would be six feet. 




PART- II. 



INDIRECT LEY ELLIN a 



CHAPTEE I. 



METHODS AND INSTRUMENTS 



(87.) Vertical Surveying. Levelling may be named Yer- 
tioal Surveying, or Up-and-down Surveying / Land Sur- 
veying being Horizontal Surveying, or Right-and-left and 
Fore-and-aft Surveying. 

All the methods of determining the position of a point in 
horizontal surveying, may be used in vertical surveying. 

The point may be determined by coordinates situated in 
a vertical plane, as in any of the systems employed in L. S. 
(Part L, Chapter I.), in a horizontal plane. 

^ „ Thus, if a balloon be held down by a sin- 

Fig. 63. 7 J 

q gle rope attached to a point in a level sur- 

face, its height above that surface is found 
by measuring the length of the rope. This 
is the Direct Method. It resembles that 
of " rectangular coordinates," L. S. (6) ; 
though here only one of the coordinates is measured. The 
other might be situated anywhere in the surface. 

If, however, the balloon be held down 
by two cords, its height can be determined 
by measuring the length of the cords and 
the distance between their lower ends. They 
correspond to the "focal coordinates" of 
L. S. (5). The required vertical height can 



Fig. 64. 




54 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



Fig. 65. 




Fig. 



Fig. 67. 



Jj? 



be calculated by trigonometry. So in the following other 
Indirect Methods. 

The length of the string of a kite, and 
the angle which this string makes with 
the horizon, are the " polar coordinates " 
of the kite ; as in L. S. (7). 

The " angles of elevation " of a me- 
teor, observed by two persons in the 
same vertical plane with it, and at known 
distances apart, are its " angular co- 
ordinates," as in L. S. (8). 

Finally, an aeronaut could determine 
his own height by observing the angles 
subtended by three given objects situ- 
ated on the earth's surface, at known 
distances, and in the same vertical plane 
with him. These angles would be the 
" trilinear coordinates " of L. S. (10). 

Many other systems of coordinate lines and angles, va- 
riously combined, may be employed. 

The desired heights may also be determined by various 
other methods, analogous to those given in L. S. for "inac- 
cessible distances." 

Combinations of measurements not in the same vertical 
plane may also be used, as will be shown in Chapter III. 

(88.) Vertical Angles. The vertical angles measured may 
be those made — either with a level 
line, or with a vertical line — by 
the line passing from one point to 
the other. 

The angle BAC is called an 
" angle of elevation," and the angle 
B'AC an "angle of depression." 
The former angle may be called pos- 
itive, and the latter negative. 

The angle BAZ or B'AZ is 




METHODS AND INSTRUMENTS. 



55 



called the zenith distance of the object. It is the complement 
of the former angle, i. e., = 90°— that angle taken with its 
proper algebraic sign. An angle of elevation, BAC = 10°, 
would be a zenith distance of 80°. An angle of depression, 
B'A C = — 10°, would be a zenith distance of 100°. The 
zenith distance is preferable in important and complicated 
operations, as avoiding the ambiguity of the other mode of 
notation. 

(89.) Instruments. All contain a divided circle, or arc, 
placed vertically, and a level or plumb line. By these is 
measured the desired vertical angle made by the inclined line 
with either a level or vertical line. 

This inclined line may be an actual line or a visual line. 
In the former case, it may be a rod, or cord, or wire, as shown 
in the figures : 

Fig. 69. 




Fig. TO. 



Fig. 71. 





This last arrangement of a cord or wire, Fig. 71, is used 
in mine surveying. A light surveyor's chain may be similarly 
used, with the advantage of giving, at the same time, differ- 
ence of heights and distance. 



56 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Diff. of lits. — length of chain x sin. angle. 
Hor. distance = length of chain x cos. angle. 

These instruments are all " Slope-measurers." They are 
also called Clinometers, Clisimeters, Eclimeters, etc., all mean- 
ing the same thing. 

(90.) Slopes, These may be designated by their angles 
with the horizon, or by the relations of their bases and heights. 
The French engineers name a slope by m& ^ 

the ratio of its height to its base; i. e., 



-r-p 5 which is the tangent of the angle 







BAC. The English and Americans use A ^f /J 

the ratio of the base to the height ; i. e., 

AC 

-p-p ? and make the height the unit, so that if A C = 2 C B, 

the slope is called 2 to 1 ; and so on. 

(91.) When the inclined line is a visual line, such as the 
line of sight of a telescope, whose angular movements are 
measured on a vertical circle beside it, and when with these 
is combined a horizontal circle for measuring horizontal an- 
gles, the instrument is called a " Theodolite." 

In the usual American form, the telescope turns over. It 
is a transit-theodolite. (See Fig. 73.) It is usually called sim- 
ply a " Transit." 

For the usual English form, see L. S., page 213. 

In the usual French form, the telescope is eccentric ; i. e., 
on one side of the vertical axis, and has a counterpoise on the 
other side, as in Fig. 141, of mining transit. 

(92.) The Surveyor's Transit, Fig. 73. The telescope re- 
volves on a horizontal axis, which itself rests on two stand- 
ards, S S, attached to the horizontal vernier-plate, H. The 
graduated vertical circle, A, by which vertical angles are 
measured, is attached to the telescope axis, and is read with a 
vernier on the lower side. A level, L, is attached to the tele- 
scope, in the same manner as that of the Y level. The ver- 



METHODS AND INSTRUMENTS. 



57 



nier-plate, which carries the telescope, is furnished with, two 
verniers on opposite sides of the instrument, and at right 
angles to the telescope. The vertical and the horizontal 
graduated circles are both furnished with a clamp and slow- 



Fig. 73. 




motion screw. Attached to the upper parallel plate is another 
clamp, C, and a pair of slow-motion screws, T T, by which 
all of the instrument above the clamp may be" given a slow 
motion, horizontally. The vernier-plate is furnished with two 
levels, at right angles to each other. One of them, D, is 



58 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

attached to the plate, and the other, E, is fastened to the 
standard, up out of the way of the second vernier. 

The compass may be used like a common surveyor's com- 
pass, the telescope taking the place of the sights. Its prin- 
cipal use is to serve as a check on the observations, the 
difference of the magnetic bearings of two lines being approx- 
imately equal to the angle measured between them by the 
more perfect instrument. 

The arrangement of the parts of the telescope, and the 
parallel plates, are the same as for the Y level. 

(93.) Adjustments. First Adjustment. To cause the bub- 
bles to remain in the centre of the tubes, when the vernier- 
plate is tarried around horizontally; i. e.,to make the plane of 
the levels perpendicular to the vertical axis of the instrument : 

To test this, turn the vernier-plate till each of the plate- 
levels is parallel to an opposite pair of the parallel plate- 
screws, and bring each bubble to the middle of its tube, by 
the screws to which it is parallel. Then turn the plate half- 
way around. If either of the bubbles runs from the centre of 
the tube, bring it half-way back, by raising or lowering one 
end of the tube, and the rest of the way, by the parallel plate- 
screws. Again, turn the plate half-way around, and repeat 
the operation, if necessary. The other tube must be tested, 
and, if necessary, adjusted in the same way. 

Second Adjustment. To cause the line of collimation to 
revolve in a plane ; i. e., to make the line of collimation per- 
pendicular to its axis : 

Set up the instrument and level it carefully. Sight to 
some well-defined point, as far off as can be distinctly seen. 

Fig. 74. - 




Clamp the instrument so that there can be no movement hori- 
zontally, turn the telescope over, and fix another point (as a nail 



METHODS AND INSTRUMENTS. 59 

driven in a stake) precisely in the line of sight, and at an 
equal distance from the instrument. 

In the figure let A be the place of the instrument and B the 
first point sighted to. If the vertical cross-hair is in adjust- 
ment, the line of sight, on turning over the telescope, will 
strike at C, A 0. being a prolongation of the straight line A B. 
If not in adjustment, it will strike on one side, as at D. JSTow 
loosen the clamp, turn the vernier-plate half-way around, and 
sight to the first object selected. Again clamp the instru- 
ment and turn over the telescope. The line of sight will now 
strike at E, as far to the right of the true line as D is to the 
left. 

To correct this, move the vertical cross-hair till the line of 
sight strikes half-way between E and C. Yerify again, and 
repeat the operation, if necessary. 

Third Adjustment. To cause the line of collimation to 
move in a truly vertical plane when the telescope is revolved ; 
i. e., to make the axis of the line of collimation parallel to the 
plane of the levels : 

Set up the instrument near the base of a spire, or other high 
object, and level it carefully. Sight to some well-defined point 
on the top of the object. Clamp the instrument so that there 
can be no motion horizontally, turn down the telescope and 
fix a point at the base of the object, precisely in the line of 
sight. ISTow loosen the clamp, turn the vernier-plate half-way 
around, and sight again to the point on the top of the object. 
Again clamp the instrument, and turn down the telescope. 
If in adjustment, the line of sight will again strike the point 
fixed at the base. If not, the apparent error is double the 
real error. Make the adjustment by raising or lowering one 
end of the axis by means of a screw, placed in the standard 
for. that purpose. 

Fourth Adjustment. To cause the line of collimation of 
the telescope to be horizontal when the bubble of the level 
attached to .it is in the centre of its tube : 

The verification and adjustment is the same as the opera- 
tion of adjusting the Y level by the " peg method," Art. (38). 



60 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

The operations of centring the object-glass and eye-piece 
are the same as for the level, Art. (37). 

Another adjustment is necessary in order that the vernier 
of the vertical circle may read zero when the bubble is in the 
centre. This is verified in various ways : 

1. By simple inspection. 

2. By reversion. Sight to some point. Note the reading 
on the vertical circle. Turn the telescope half-way around 
horizontally. Turn over the telescope and again observe the 
same point, and note the reading. Half the difference (if 
any) of the two readings is the error. 

The principle is that given in L. S. (334). 
This method requires the instrument to be a transit- 
theodolite. 

3. By reciprocal observations. Observe successively from 
each of two points to the other. Half the difference of the 
readings equals the index-error. 

When the verification has been made, the error may be 
rectified on the instrument, or noted as a correction to each 
observation, when the instrument is large and delicate. 

(94.) Field-Work. . To measure horizontal angles. Set the 
transit so that its centre shall be precisely over the angular 
point. This is done by means of a .plumb-line, suspended 
from the centre of the instrument. Level the instrument 
carefully. Sight to a rod, held at some point on one of the 
lines, as at B in the figure (A being the place of the transit), 
and note the reading. Then loosen the clamp of the vernier- 
plate, keeping the other plate clamped ; sight to a rod held at 
some point on the second line, as at C, and again note the 

Fig. 75. 



reading. The difference of the two readings will give the an- 
gle B A C. This is the angle of intersection. 



METHODS AND INSTRUMENTS. 61 

To measure the angle of deflection, D A C, i. a, the angle 
between A and B A prolonged : After sighting to B, turn 
over the telescope. It will now point toward D, in the line 
B A prolonged. Note the reading, sight to 0, and again note 
the reading. The difference of the readings will give the re- 
quired angle. 

Vertical angles are measured similarly to horizontal ones, 
only using the vertical instead of the horizontal circle. 

Traversing. In this method of surveying and recording a 
line, the direction of each successive portion is determined, 
not by the angle which it makes with the line preceding it, 
but with the first line observed, or some other constant line. 
The operation consists essentially in taking each back-sight 
by the lower motion (which turns the circle without changing 
the reading), and taking each forward sight by the upper mo- 
tion, which moves the vernier over the arc measuring the new 
angle ; and thus adds it to or subtracts it from the previous 
reading. 

Fl6 - TO - , Set up the instrument 

o <*^-j— - — --^iL..-~ a t some station, as B ; 

put the vernier at zero, 
and, by the lower mo- 
tion, sight back to A. 
Tighten the lower clamp, 
reverse the telescope, loosen the upper clamp, sight to by 
the upper motion, and clamp the vernier-plate again. Re- 
move the instrument to C, sight back to B by the lower motion. 
Then clamp below, reverse the telescope, loosen the upper 
clamp, and sight to D by the upper motion. Then go to D 
and proceed as at C ; and so on. The reading gives the an- 
gles measured to the right or " with the sun," as shown by 
the arcs in the figure. 

(95.) Angular Profiles. A section or profile of a tolerably 
uniform slope is most easily obtained, as shown in the figures, 
by measuring the heights or depths below an inclined line, 
instead of below a level line. 




62 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Fig. 77. 




Fig. 78. 




A cross-section for a road may be taken in the same way. 

(96.) Burnier's Level. It is a pear-shaped instrument, hav- 
ing two graduated circles; one vertical, having a weight 
attached so as to keep it in the 
same vertical position when in 
use; and the other, a horizontal 
graduated circle, made light and 
carried around by a magnetic nee- 
dle, so that the instrument can be 
used as a compass as well as a 
slope or angular level. It has a 

convex-glass, or lens, in the smaller end, through which can 
be seen a hair which covers, on the circle, the number of the 
degrees of the angle of inclination, or of the horizontal angle. 

The sights are on the top or sides, according as it is used 
as a compass or slope-measurer. It is used by sighting to the 
object, and at the same time reading off the angle, the hair 
covering the zero-mark when the instrument is level. 

(97.) German Universal Instrument. Its use is to enable the 
observer to sight to an object nearly or 
quite overhead. It consists of a tele- 
scope having the part which carries the 
eye-piece at right angles to the part 
carrying the object-glass, instead of be- 
ing in the same straight line, as in an 



Fig. 79. 



3*~Eg j- -~jj f a 



SIMPLE ANGULAR LEVELLING. 



63 



ordinary telescope. The part containing the eye-piece is con- 
nected with the other part at the axis, and is in the same line 
with the axis. 

In the telescope is placed a small mirror, or reflector, or 
(what is still better) a triangular prism of glass, at an angle 
of 45° to the line of sight. Thus the observer can keep his 
eye at the same place at any inclination of the telescope. 



CHAPTEK II. 



SIMPLE AKGULAE LETELLIIG. 



A. — For Short Distances. 



Fig. 80. 



A y 




' 




(98.) Principle. For short dis- 
tances, curvature and refraction 
may be neglected. Thus, if the 
height of a wall, house, tree, etc., 
be desired, note the point where the 
horizontal line strikes the wall, etc., 
and add its height above the ground 
to that calculated by the formula : 



BC 



A C. tang. BAC. 



[1-] 



(99.) The "best-condition" angle for observation (see L. S., 
383) is 45°. Hence, in setting the instrument, we should, 
where practicable, have the distance about equal to the height 
of the point whose height we wish to ascertain. 



64 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



B. — For Greater Distances. 

(100.) Correction for Curvature. A C is 

the line of apparent level, as given by the 
instrument, and A C is the line of true 
level. Calling the angle A C B = 90° 
(which it is approximately for moderately 
great distances), formula [1] gives B C as 
the height of B above A. But B C is the 
true difference of heights of A and B. 

A correction for the curvature of the 
earth must therefore be made. It may 
be done in two ways : either by calcu- 
lating C C, and adding it to B C, obtained 
by formula [1] ; or by calculating the an- 
gle C A C, adding it to B A C, and then 
applying the formula [1] to the angle 
B AC 







u\ 



(101.) Correcting the Result. Expressing the distance by k, 
we have, by (14) : 



InfeetCC -A=2x 2^888629 



0.000000023936^. 



Then, calling A C B a right angle, we have : 

BC = k x tang. B A C + 0.000000023936& 3 in ft. [2.] 

The arc A C and the straight lines A C and A C are all 
three approximately equal. 

(102.) Correcting the Angle. The angle CAC^JAOC, 
the central angle, which is measured by the arc A C or k. 

The length of the arc subtending one minute 
2tt x 20888629 



= 6076 ft. 






cu-K^o**- 



--H^ik. 



360 x 60 



U. .- 



jujC4 4L^ ***™> 






SIMPLE ANGULAR LEVELLING. 

Then, for any arc, Jc, the angle O in minutes 
i 



65 



6076 



= 0.0001646& 



and the angle CAC ; (in minutes) = 0.0000823&. 

Adding this to the observed angle, BAC, and calling 
AC'Ba right angle, we have, by [1] : 



B C = Jg tang. (BAC + 0.0000823&). 



[3.] 



Fig. 82. 



(103.) Correction for Refraction. The effect of refraction 

causes the angle actually ob- 
served to be, not CAB, but 
C A B', which will be desig- 
nated by a°. For small dis- 
tances, B and B' sensibly co- 
incide. The correction for 
refraction may be made in 
two ways, as for curvature. 

To correct the result by 
finding B B'. It varies very 
irregularly, with wind, ba- 
rometer, temperature, etc. ; 
but is usually taken, as an 
average, BB^ 0.16 C C. * 

Subtracting this from the value of B C, in formula [2], it 
becomes BC^L tang. B'A C + 0.00000002& 2 . [4.] 

To correct the observed angle. Subtract from it the angle 
B A B', which is about 0.16 of the angle C A C 

This changes formula [3] to 

BC' = Jc. tang. (B' A C + 0.000069&). [5.] 

C. — For Very Great Distances. 

(104.) Correction for Curvature. As before, there are two 
methods of making the correction. 







aJT'C' &V 



Q6 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

For these distances we cannot consider the angle at C' a 
right angle. The triangle ABC gives 

sin. BAG 
B C = h . — = — tj— • 
sin. B 

To find the angle B, we have, in the triangle BAO, 

B = 180° -(O + BAO), 

B = 180° - (O + 90° + B AC), 

B = 90° - (0 + B A C) ; 

Hence, sin. B = cos. (O + B A C). 

sin. B A C 



Then, BC = ^ 



cos. (O + BAC)' 



and B C'= B C + C C'= h. — - ^ , -p A n< + 0.000000023936/P 

cos. (O + BAC) 

B C'=h ~-S^i ^ _^ + 0.000000023936& a . [6.1 

cos. (B A C + 0.0001646&) L J 

Correcting the Angle. In the triangle A B C, getting 
expressions for the angles, and using the sine proportion, as 
before, in A B C, we have : 

snMBAC+iO). 
^ -*-co S . (BAC+O) 

R „_, sin.(BA C+0.000082 3&) 

~ cos.(BAC+0.0001646£) L '' J 

(105.) Correction for Refraction. Formula [6] becomes 

sin^B'AO-OOOOOlSie^) 
B ° - %os. (B'A C + 0.00015143/fc) + - 00000002393t,A " ^ 



SIMPLE ANGULAR LEVELLING. 



67 



Formula [7] becomes, diminishing B A C in both numer- 
ator and denominator by 0.08 of O, 



BC^^. 



sin. (B r A C + 0.00006913^) 
cos. (B' A C + 0.00015143^) 



m 



Fig. 83. 



(106.) Reciprocal Observations for Cancelling Refraction. Ob- 
serve the reciprocal angles of eleva- 
tion and depression from each point 
to the other. Call these angles 
a and (3. Then : 




BC' = & 



sin. i (a 4- P) 



[10.] 



cos. $(a + j3 + 0) 

Note.— Angle O, in minutes — 0.0001 646&. 
Log. 0.0001646 = T.2164298. 

When zenith distances are ob- 
served, they are denoted by d and <5 ; . 
Then formula [10] becomes : 



cos. i(& — <5+ O) 



[10'.] 



"When O is very small, compared with the other angles, by 
neglecting it we have : 



BC'^L tang, i (a + /3). 
Using zenith distances, this becomes : 



[11.] 



[11'.] 



(107.) Reduction to the Summits of the Signals. Stations a 
and h cannot be seen from one another. Signals, Art. (240), are 
erected at each point, and from a the angle B a C = A is 
observed ; and from h the angle A h D = B. The heights of 
the signals above the instrument are h and h\ 



68 



LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



Required the reduced angles a and (3. 



Fig. 84. 



a = A — 



P = B + 



h . cos. A 
h. sin. 1" 

h! . cos. B 
A . sin. V 



[12]. 




The difference is in seconds. 

Usually, in such cases, zenith dis- 
tances are taken, and the observed an- 
gles are called A and A r . The re- 
duced angles are 6 and 6'. Then the formulas become 

h . sin. A - ., „. , ti. sin. A ' 

d= A + 7 . „,_ -,,/ , and d'= a' + 



& . sin. 1' 



^.sin.r- 



[13.] 



The difference is in seconds. 

Instead of h and N some writers use dH. and dW ; or dA 
and <£A/, meaning difference of height, and difference of alti- 
tude. 

For great exactness, instead of using the mean radius of 
the earth to get O, the radius at the point of observation 
is used. 

(108.) When the height of the signal above the instrument 
cannot be measured, if the signal be conical, like a spire, etc., 

Fig. 85. 




to find B B' we measure two diameters, 2 E and 2 r, and the 
distance apart, h. 



SIMPLE ANGULAR LEVELLING. 

Then, B B' = 



69 



B-r" & 4 -] 

If the oblique distance I be measured instead of A, then 

E 

B^ [* + (B -*■)] [J -(B -*•)]. [1.5.] 



BB' = 



When the spire is very aeute, then this method is inac- 
curate. , 

Fig. 86. 

IB 




. Take some point, A, and observe zenith distances, 6, 6 f \ 
and d'. Then : 



BB' 



h. tang. {d» - 6) 
cos.i(cf / -d + 6} 



[1.6.] 



(109.) Levelling by the Horizon of the Sea. From an emi- 
nence, as B, sight to the sea horizon, and measure 6°= angle 
ABZ. Then: 

Fig. 87. 

Z/ 




BC -* e (i%^)V-9ot [i+i (r^)V- 90 °)i- f 17 -] 

(6° — 90°) is to be reduced to seconds. It is equal to the 
angle of depression at B ; n is the coefficient of refraction. It 
is taken at 0.08. 



70 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



CHAPTER III. 



COMPOUND ANGULAR LEVELLING 



The following problems may mostly be reduced to a com- 
bination of: first, determining the inaccessible distance to a 
point immediately nnder (or over) the point whose height is 
desired, and then using this distance to obtain that height. 



Fig. 88. 




(110.) By Angular Co-ordinates in one Plane. Take two sta- 
tions, A and D, in the same vertical plane with B. At A 
observe the angles of elevation of B and D. Measure A D. 
At D observe augle ADB. Then, in triangle A B D we get 
A B, and in triangle B A C we get B C. 



bc = ad; 



sin. BDA.sin.BAC 



sin. A B D 



[18.] 



Eig. 89. 



For great distances, the corrections for curvature and re- 
fraction are to be made as in last chapter. 

If AD be horizontal, the same- formula 
applies ; but there is one angle less to meas- 
ure ; since BAG = BAD. Formula [18] 
gives the height of B above A. 

If the height of B above D, in Fig. 88, be 
desired, find B D in the triangle BAD, observe the angle of 
elevation of B from D, and then the desired height equals 

B D. sin. B D E. 




COMPOUND ANGULAR LEVELLING. 



71 



Otherwise, find height of D above A, and subtract it 
from B C. 

(111.) By Angular Co-ordinates in several Planes. On'irreg- 
ular ground, when the distance between the two points is 
unknown, the operations for finding it by the various methods 
of L. S., Part VIL, Chapter III., may be combined with the 
observation of vertical angles, thus : 

Fig. 90. 




At A measure the vertical angle of elevation, B A C. Also 
measure the horizontal angle, C A D, to some point, D, and 
measure horizontally the distance, A D. At D measure the 
horizontal angle ADC. Then : 



AC = AD 



sin. ADC 
sin. A C D ' 



B C = A C . tang. B A C 



t> n — a -r sm * ADC. tan g- B A C 
sm. A C D 



[19.] 



Fig. 91. 




(112.) Conversely. The distance 
may be obtained when the height 
is known. 

Let C B be a known height : 
Then, AC= CB. tan. ABC. 



72 LEVELLING, TOPOGKAPHY, AND HIGHER SURVEYING. 



B C is a known height, and D E an 
inaccessible line in the same horizontal B 
plane as C. Find CD and C E by 
the last method, and measure the hor- 
izontal angle ECD snbtended at C by 
ED. 

Then two sides and the included 



angle of a triangle are known, to find the third side. 




PART III. 
BAROMETRIC LEVELLING 



CHAPTER I. 

PRINCIPLES AND FORMULAS. 

(113.) Principles. The difference of the heights of two 
places may be determined by finding the difference of their 
depths below the top of the atmosphere in the same way as 
the comparative heights of ground under water are determined 
by the difference of the depths below the top of the water. 
The desired height of the atmosphere above any point, such 
as the top of a mountain, or the bottom of a valley > is deter- 
mined by weighing it. This is done by trying how high a 
column of mercury or other liquid the column of air above it 
will balance ; or what pressure it will exert against an elas- 
tic box containing a vacuum, etc., etc. Such instruments are 
called Barometers. 

(114.) Applications. Since the column of mercury in the 
barometer is supported by the column of air above it, the 
mercury sinks when the barometer is carried higher, and vice 
versa. 

The weight of any portion of air decreases from the surface 
of the earth to the assumed surface of the atmosphere. It has 
been found that, as the heights to which the barometer is car- 
ried increase in arithmetical progression, the weights of the 
column of air above the barometer, and consequently its read- 



LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

ings, decrease in geometrical progression. Consequently, the 
difference of the heights of any two, not very distant, points 
on the earth's surface, is proportional to the difference of the 
logarithms of the readings of the barometer at those points ; 
i. e., equal to this latter difference multiplied by some constant 
coefficient. This is found by experiment to be 60159, at the 
freezing point, or temperature of 32° F., the readings of the 
mercury being in inches, and the product, which is the differ- 
ence of height, being in feet. 

Several corrections are necessary. 

(115.) Correction for Temperature of the Mercury. If the 

temperature of the mercury be different at the two sta- 
tions, it is expanded at the one, and contracted at the other, 
to a height different from that which is due to the mere 
weight of the air above it. 

Mercury expands about 10 S 00 of its bulk for each degree 
of F. Therefore, this fraction of the height of the column is 
to be added to the height of the colder column, or subtracted 
from the height of the warmer one, in order to reduce them 
to the same standard. A thermometer is therefore attached 
to the instrument in such a manner as to give the temperature 
of the mercury. 

If a brass scale is used, the correction is 1 9 for each 
degree F. 

(116.) Correction for Temperature of the Air. The warmer 
the air is, the lighter it is ; so that a column of warm air of 
any height will weigh less than when it is colder. Con- 
sequently, the mercury in warm air falls less in ascending any 
height, and is higher at the place than it otherwise would be. 
Hence the height given by the preceding approximate result 
will be too small, and must be increased by -^^ part for each 
degree F. that the temperature of the air is above 32°. The 
effect of moisture in the air changes this fraction to ^-g-. 



(117.) Other Corrections. For very great accuracy, we 
should allow for the variation of gravity, corresponding to the 



PRINCIPLES AND FORMULAS. 



75 



variation of latitude on either side of the mean. So, too, we 
should allow for the decrease of gravity corresponding to any 
increase of height of the place. 

(118.) Rules for Calculating Heights by the Mercurial Ba- 
rometer. 1. At each station read the barometer; note its 
temperature by the attached thermometer, and note the tem- 
perature of the air by a detached thermometer. 

2. Multiply the height of the upper column by the differ- 
ence of readings of the attached thermometer, and that by 
To~oWu> and add the product to the upper column, if that be 
the colder, or subtract it, if that be the warmer. This gives 
the corrected height of the mercury. 

3. Multiply the difference of the logarithms of the cor- 
rected heights of the mercury (i. e., the corrected upper one 
and the lower one) by 60159, and the product is the approxi- 
mate difference of heights of the places in feet for the temper- 
ature of 32°. 

4. Subtract 32° from the arithmetical mean of the temper- 
atures of the detached thermometer; multiply, the approxi- 
mate altitude by this difference ; divide the product by 450 ; 
add the quotient to the approximate altitude, and the sum is 
the corrected altitude. 



(119.) Formulas. 

in formulas, thus : 



The rules just given are best expressed 



Height of Mercury 


At Lower Station. 


At Upper Station. 


H 
T 
t 


* h' 
T' 

a 


Temperature of Mercury 

Temperature of Air 





Calling the reduced height of mercury at upper station A, 
we have, by Eule 2 : 



A = A' + 0.00009(T-T')A'. 



[1.] 



(K B. If T is more than T, the product will be sub- 
tractive.) 



76 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Then, by Eule 3, we have : 

Approx. height = 60159 (log. H — log. A). 
By Eule 4, the correction for temperature of air 

t + f — 64: 



= approx. height x 



900 



Adding this correction to the approximate height, and fac- 
toring the sum, we get : 

Corrected ht. = 60159 (log. H - log. h) (l + + 9Q ~ \ [2]. 

(120.) To Correct for Latitude. Multiply the preceding re- 
sult by 0.00265 . cos. 2L (L being the latitude), and add (alge- 
braically) the product to the preceding result. 

At 45°, correction is zero. At equator it is + 0.00265. At 
pole it is —0.00265. 

To Correct for Elevation of the Place. Call the last cor- 
rected height a/, and the height of the lower place above the 
level of the sea S, and add to x' this quantity : 

x f + 52251 S 

+ 



20888629 ' 10444315 

(121.) Final English Formula. Combining the previous re- 
sults into one formula, we get : 

t+t f - 64 ^ 



V 1 + 900 P 



Ht. = 60159(Log.H-log.A) <(1 + 0.00265. cos. 2 L), 



(i+ 



a/+ 52251 S 



+ 



20888629 ' 10444315 



[3]. 
In this formula, the three quantities under each other are 
three factors. 



PRINCIPLES AND FORMULAS. 77 

Usually, only the first factor is required, and then we have 
formula [2]. Using the second also we correct for latitude ; 
and using the third, for the elevation. 

(122.) French Formulas. French barometers are graduated 
in French millimetres, each = 0.03937 inch., and the ther- 
mometer is centigrade, in which the freezing-point is zero, 
and boiling-point 100° : 

a°Cent. = (§■ a + 32)° F. 

Then, the French formula corresponding to [3] is the fol- 
lowing (H and N being in millimetres, and the temperatures 
centigrade) : 

A = A '( 1 + w)- 

And the difference of heights in metres 

v 1 + "moo r 

- lo g- h ) ( ( 1 + 0.00265 . cos. 2 L), ) w 

/ x ' + 15 926 S x 

V + 6366198 + 3183099,) 

(123.) Babinet's Simplified Formula, without logarithms. 

/ T — T\ 

h! is reduced to A, as before, viz. : h = h! ( 1 + fi nn j • 

Then, the difference of heights in metres 



= 16000. 



H- h (^ 2(t+ t') 
H + h 



(• + l Uf 2 > w 



The heights are in millimetres and the temperatures centi- 
grade. 



78 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Example. H = 755. h = 745 
. t = 15° tf = 10° 



10 / 50 \ 

1500 V + 1000/ 



Ht. = 16000 -^ 1 + --- = 112 m. 



Correct result is 111.6 m 

This formula is a very near approximation for moderate 
heights. 

Babinet's formula in English measures (the heights being 
in inches, and temperatures Fahrenheit) is in feet : 

52494 






/H - A\ / ^ + ^_64\ 



H + A. 

(124.) Tables. These. shorten the operations greatly. The 
most portable are in " Annuaire du Bureau des Longitudes." 
The most complete are Prof. Guyot's, published by the Smith- 
sonian Institute at Washington. 

(125.) Approximations. -^ of an inch difference of read- 
ings at two places corresponds to about 90 feet difference of 
elevation. 1 millimetre difference of readings corresponds to 
about 10|- metres difference of height, or about 34 feet. 

This is correct near the freezing-point, and near the level 
of the sea. The height corresponding to any given difference 
of readings increases, however, with the temperature and with 
the height of the station. Thus, at 70° F., ^ of an inch cor- 
responds to an elevation of 95 feet ; and 1 mm. at 30° Cent, 
corresponds to llf metres, or about 40 feet. 



INSTRUMENTS. 79 

CHAPTEE II. 

INSTRUMENTS. 

(126.) Barometers made for levelling are called Mountain 
Barometers. They are either cistern barometers or siphon 
barometers.' The best of the former is Fortin's, as improved 
by Prof. Guyot. (See Fig. 93.) This consists of a col- 
"'** umn of mercury contained in a glass tube, whose lower 
end is placed in a cistern of mercury. The tube is cov- 
ered with a brass case, terminating at the top in a ring, 
A, for suspension, and at the bottom in a flange, B, to 
which the cistern is attached. 

At C is a vernier by which the height of the mercury 
is read off. The zero of the scale is a small ivory point, 
P. The mercury in the cistern is raised or low- FlG 94 
ered, by means of the milled-headed screw O, till 
its surface is just in contact with the ivory point. 
At E is the attached thermometer which indi- 
cates the temperature of the mercury. When it 
is carried, the mercury is screwed up to prevent 
breaking the glass. 1 

In the siphon barometer, the cistern is dis- 
pensed with. The tube is turned up at the 
lower end, as shown in Fig. 94, and a small hole, 
at T, admits the air. The difference of heights 
of the mercury in the two branches of the tube 
is taken as the height of the mercurial column. 
It is enclosed in a brass case, and furnished with ver- 
niers, thermometers, etc., as in the preceding form. It 
is carried inverted, to avoid breaking. 

The best is Gay-Lussac's, improved by Bunten. 

1 For a complete description, see Tenth Annual Report of Smithsonian Insti- 
tute. 



80 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



(127.) The Aneroid Barometer. This is a thin box of corru- 
gated copper, exhausted of air. When the air grows heavier, 
the box is compressed ; and when the air grows lighter it is 

Fig. 95 




expanded by a spring inside. This motion is communicated 
by suitable levers to the index-hand, on the face, which indi- 
cates the pressure of the atmosphere, the face being graduated 
to correspond with a common barometer. 

It is much used on account of its portability, but is not as 
reliable as the mercurial barometer. 

(128.) The temperature at 
which water boils varies with 
the pressure of the atmos- 
phere, and therefore decreas- 
es in ascending heights. Then 
a thermometer becomes a sub- 
stitute for a barometer. 

Approximately, each de- 



Temperature 


Corresponding 


of Boiling Water. 


Barometer Readings. 


213° 


30".522 


212° 


29".922 


211° 


29".331 


210° 


28". 751 


209° 


28".180 


208° 


27".618 



INSTRUMENTS. 



81 



gree of difference (Falir.) corresponds to about 550 feet . differ- 
ence of elevation, subject to the usual barometric corrections 
for the temperature of the air. For minute tables, see Guyot's.. 

(129.) Accuracy of Barometric Observations. This increases 
with the number of repetitions of them, the mean being 
taken. With great skill and experience they may be depended 
upon to a very few feet. 

PBOFESSOE GUYOT'S EESULTS. 



Heights found by the 
Barometer. 


Corresponding 

Heights found by the 

Spirit-Level. 


6707 feet. 
2752 " 
6291 " 


6711 feet. 

2752 " 
(6285 " 
( 6293 " 



(130.) The observations at the two places, whose difference 
of heights is to be determined, should be taken simultaneously 
at a series of intervals previously agreed upon, the distance 
apart of the places being as short as possible. Distant places 
should be connected by a series of intermediate ones. 



PART IV. 
TOPOGRAPHY. 



INTRODUCTION. 



(131.) Definition. Topography is the complete determina- 
tion and representation of any portion of the surface of the 
earth, embracing the relative position and heights of its ine- 
qualities ; its hills and hollows, its ridges and valleys, level 
plains, slopes, etc., telling precisely where any point is, and 
how high it is. 

It therefore determines the three coordinates of any point; 
the horizontal ones by surveying, and the vertical one by 
levelling. 

The results of these determinations are represented in a 
conventional manner, which is called "topographical map- 
ping." 

The difficulty is, that we see hills and hollows in elevation, 
while we have to represent them in plan. 

(132.) Systems. Hills are represented by various systems : 

1. By level contour-lines, or horizontal sections. 

2. By lines of greatest slope, perpendicular to the former. 

3. By shades from vertical light. 

4. By shades from oblique light. 

The most usual method is a combination of the first, second, 
and third systems. 



FIRST SYSTEM. 



83 



CHAPTER I. 

FIRST SYSTEM. 

BY HORIZONTAL CONTOUR-LINES. 

(133.) General Ideas. Imagine a hill to be sliced off by a 
number of equidistant horizontal planes, and their intersec- 
tions with it to be drawn as they would be seen from above, 
or horizontally projected on the map, as in Fig. 96. These 
are "contour-lines." 

Fig. 96. 




They are the same lines as would be formed by water sur- 
rounding the hill, and rising one foot at a time (or any other 
height), till it reached the top of the hill. The edge of the 



Fig. 97. 



Fig. 98. 



Fig. 99. 





84 LEVELLING, TOPOGKAPHY, AND HIGHER SURVEYING. 

water, or its shore, at each successive rise, would be one of 
these horizontal contour-lines. It is plain that their nearness 
or distance on the map would indicate the steepness or gentle- 
ness of the slopes. A right cone would thus be represented 
by a series of concentric circles, as in Fig. 97 ; an oblique cone, 
by circles not concentric, bat nearer to each other on the steep 
side than on the other, as in Fig. 98 ; and b^a half-egg, some- 
what as in Fig. 99. 

(134.) Plane of Reference. The horizontal plane on which 
the contour-lines are projected, and to which they are re- 
ferred, is called the " plane of reference." This plane may be 
assumed in any position, and the distance of the contour-lines 
above or below it is noted on them. It is usually best to 
assume the position of the plane of reference lower than any 
point to be represented ; so that all the contour -lines will be 
above it, and none of them have minus signs. (See Art. 59.) 

(135.) Vertical Distances of the Horizontal Sections. These 
depend on the object of the survey, the population of the coun- 
try, the irregularity of the surface, and the scale of the map. 
In mountainous districts they may be 100 feet apart. On the 
United States Coast Survey they are 20 feet. For engineering 
purposes, 5 feet, or less. One rule is to make the distance in 
feet equal to the denominator of the ratio of the scale of the 
map, divided by 600. 

(136.) Methods for determining Contour-Lines. They are of 
two classes : 1. Determining them on the ground at once ; 2. 
Determining the highest and lowest points, and thence de- 
ducing the contour-lines. 

First Method. 

(137.) General Method. Determine one point at the desired 
height of one line, as in Art. (81) ; and then "locate " a line 
at that level, as in Art. (84). 

The "reflected hand-level," or " reflecting-level," or "wa- 



FIKST SYSTEM. 



85 



ter-level," are sufficiently accurate between "bench-marks" 
not very distant. 

One such line having been determined, a point in the next 
higher or the next lower one is fixed, and the preceding oper- 
ations repeated. 

(138.) On a long, narrow Strip of Ground, such as that re- 
quired for locating a road: Run a section across it at every \ 
or -J mile, about in the line of greatest slope. Set stakes on 
these sections at the heights of the desired contour-lines, 
and then get intermediate points at these heights between the 
stakes. These sections check the levels. 

Fig. 100. 




(139.) On a Broad Surface. Level around it setting-stakes, 
at points of the desired height, and then run sections across it, 
and from them obtain the contour-lines as before. 

The external lines here serve as checks to the cross-lines. 



(140.) Surveying the Contour-Lines. The contour-lines thus 
found may be surveyed by any method. If they are long, and 
not very much curved, the compass and chain and the method 
of " progression " is best. (See L. S. 220.) If they are curved 
irregularly, the method of radiation is best. When straight 
lines exist among them, such as fences, etc., or can conveniently 
be established, then rectangular coordinates are most con- 
venient. 



86 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(141.) Contouring with the Plane-Table. 1 It is used to map 
the points as soon as obtained, thus : Range out an approxi- 
mately level line, and on it set equidistant stakes. At these 
stakes range out perpendiculars to the line, and set up several 
stakes on them for the alignment of the rodman. Draw these 
lines on the plane-table. Set up and "orient" (L. S. 456) 
the table on the ground. Send the rod along one of the per- 
pendiculars till it comes to a point of the right height. Then 
sight to it with the alidade, and its edge will cut the corre- 
sponding line on the table at the correct place on the plat. So 
for the other perpendiculars. 

Second Method. 

(142.) General Nature, This method consists in determin- 
ing the heights and positions of the principal points, where 
the surface of the ground changes its slope in degree or in 
direction, i. e., determining all the highest and lowest points 
and lines, the tops of the hills and bottoms of the hollows, 
ridges and valleys, etc., and then, by proportion or interpola- 
tion, obtaining the places of the points which are at the same 
desired level. The heights of the principal points are found 
by common levelling, and their places fixed as in Art. (141). 

The first method is more accurate. The second is more 
rapid. 

(143.) Irregular Ground. When the ground has no very 
marked features, run lines across it in various directions, and 
level along them, taking heights at each change of slope, just 
as in taking sections for profiles. 

Otherwise, thus : Set stakes on four sides of the field, so 
as to enclose it in a rectangle, if possible, as in Fig. 101. 
Place the stakes equidistant, so that the imaginary visual 
lines connecting them would divide the surface into rectangles. 
Send the rod along one of these lines till it gets in the range 
of a cross-one, and observe to it there. Put down the ob- 
served heights of these points at the corresponding points on 

1 For description and method of using the Plane-table, see L. S. Part VIII. 



FIRST SYSTEM. 



the plat, on which these lines have been drawn, 
tour-lines are determined as in Art. (146). 



87 
The con* 



Fig. 101. 



i ! 

'. .(. __J_ 



(144.) On a Single Hill. Proceed thus : From its top, range 
lines down the hill, in various directions, and take their bear- 
ings. Set stakes on them at each change of slope, and note 
the heights and distances of these stakes from the starting- 
point, and plat their places. The contour-lines are then put 
in as in Art. (146). 

With a transit, the heights of the points could be deter- 
mined by vertical angles ; and also their distances with stadia- 
hairs, their directions being given by the horizontal circle of 
the transit. The French use for this purpose a " levelling 
compass." 

(145.) For an Extensive Topographical Survey. Proceed thus : 
Set up, and get the height of the cross-hairs from. some bench- 
mark, and get the heights of high and low prominent points 
all around. Then go beyond these points and set up again. 
Sight to one of these known points as a " turning-point," and 
get the heights of all the points now in sight, as before. Then 
go beyond these again, and so on. The places of these new 
points are fixed as before. 



88 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(146.) Interpolation, The heights and the places of the 
principal points being determined, by either of the pre- 



50.00 



Fig. 102. 

45.00 40.00 





36.00 ^ 


33 00 


35.00 


-^_____ 


___^^-""'~ 


^ 


■ 




\^ 






-___ 


, 







35.00 



ceding methods, points of any intermediate height, corre- 
sponding to any desired contonr-curve, are obtained by pro- 
portion. 

If, in Fig. 102, the heights of the intersection of the lines 
being found, as in Art. (143), and their distance apart being 
100 feet, it is required to construct contour-curves whose dif- 
ference of heights is 5 feet: Taking for example the one 
whose height is 45 feet, we see it must fall between the points 
A and B, whose heights are 50 feet and 35 feet; and its dis- 
tance from A will be found by the proportion, as 15 is to 5 so 
is 100 to the required distance. So on for any number of 
points. To save the labor of continually calculating the fourth 
proportional, a scale of proportion may be constructed. 

(147.) Interpolating with the Sector. (L. S. 52.) This is 
the easiest way. The problem is : having given on a plat two 
points of known height, to interpolate between them a point 
of any desired intermediate height. 

Take in the dividers the distance between the given points 
on the plat ; open the sector so that this distance shall just 



FIRST SYSTEM. 



89 



Fig. 103. 



reach between numbers, on the scale 
marked L, corresponding to the dif- 
ference of the heights of the two 
given points ; i. e., from 6 to 6, or 
7 to 7, and so on. The sector is 
then set for all the interpolations 
between these two points. 

Then note the difference of 
height between the desired point 
and one of the given points, and ex- 
tend the dividers between the cor- 
responding numbers on the scale. 
This opening will be the distance 
to be set off on the plat from the given point to the desired 
point. 




(148.) Ridges and Thalwegs. The general character of the 
surface of a country is given by two sets of lines : the ridge- 
lines, or water-shed lines / and the " thalwegs" or lowest lines 
of valleys. 

. The former are lines which divide the water falling upon 
them, and from which it passes off on contrary sides. They 
are the lines of least slope when looking along them from 
above downward ; and they are the lines of greatest slope 
when looking from below upward. They can therefore be 
readDy determined by the slope-level, etc. They are the 
lines of least zenith distances when viewed from either direc- 
tion. 

On these lines are found all the projecting or protruding 
bends of the contour-lines, convex toward the lower ground, 
as shown in Fig. 104. 

The second set of lines, or the " thalwegs," are the con- 
verse of the former. They are indicated by the water-courses 
which follow them or occupy them. They are the lines of 
greatest slope when looked at from above, and of least slope 
when looked at from below. They are the lines of greatest 
zenith distance when viewed from either direction. 



90 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



On these lines are the receding or reentering points of the 
eontour-curves, concave toward the lower gronnd. 



Fig. 104. 




The general system of the surface of a country is most 
easily characterized by putting down these two sets of lines, 
and marking the changes of slope, especially the beginning 
and the end. 

The most important points to be determined are : 

1. At the top and bottom of slopes. 

2. At the changes of slopes in degree. 

3. On the water-shed lines, and on the thalwegs. 

4. On " cols," or culminating points of passes. 

(149.) Forms of Ground. It will be found on the iuspection 
of a " contour-map " (which shows 
ground much more plainly to the 
eye than does the ground itself), that 
its infinite variety of form may, for 
the purposes of the engineer, be re- 
duced to five : 1. Sloping down on 



Fig. 105. 




all sides ; i. e., a hill, Fig. 105. 



Fig. 106. 



Fig. 107. 





FIRST SYSTEM. 



91 



2. Sloping up on all sides ; i. e., a hollow, Fig. 106. 

3. Sloping down on three sides and up on one ; i. e., a 
croupe, or shoulder, or promontory, the" end of a ridge or 
water-shed line, Fig. 107. 

4. Sloping up on three sides and down on one ; i. e., a 
valley, or " thalweg," Fig. 108. 

5. Sloping up on two sides and sloping down on two, al- 
ternately ; i. e., a " pass," or " col? or " saddle," Fig. 109. 



Fig. 108. 





[Note. — The arrows in the figures indicate the direction in which water 
would run.] 

(150.) Sketching Ground by Contours. A valuable guide is, 
the observation that the lines are perpendicular to the water- 
shed lines and thalwegs. ISTote especially the contour-lines at 
the bottoms of hills and ridges, and at the tops of hollows and 
valleys, putting them down, in their true relative positions and 
distances, to an estimated scale. 

On a long slope or hill, draw first the bottom contour-line, 
and the top one ; and then the middle one ; and afterward 
interpolate others. Remember that two of them can never 
meet, except on a perpendicular face ; and that, if one of them 
passes entirely around a hill or hollow, it will come back to 
its starting-point. Hold the field-book so that the lines on it 
have their true direction. 



(151.) Ambiguity. In contour-maps of ground, if the 
heights of the contour-lines are not written upon them, it may 



92 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

be doubtful which are the highest and lowest; which are 
ridges and which valleys, etc. 

1. Numbers remove this. 

2. The water-courses show the slopes. It' there are none, 
put some in, in the thalwegs of a rough sketch. 

3. Put hatchings on the lower sides of the contour-lines, as 
if water were draining off. 

4. Tint the valleys and low places. 

(152.) Conventionalities. Sometimes the spaces between 
contour-lines are colored with tints of Indian-ink, sepia, etc., 
increasing in darkness as the depth increases. 

Ground under water is commonly so represented, begin- 
ning at the low- water line and covering the space to the six- 
feet-deep contour-line with a dark shade of Indian-ink ; then a 
lighter shade from 6 to 12; a still lighter from 12 to 18, and 
the lightest from 18 to 24. 

Greater depths are noted in fathoms and fractions. 

(153,) Applications of Contour-Lines. They have many im- 
portant uses besides their representation of ground : 

1. To obtain vertical sections ; i. e., profiles. 

2. To obtain oblique sections. 

3. To locate roads. 

4. To calculate excavation and embankment. Consider 
the contour-lines to represent sections of the mass by horizon- 
tal planes. Then each slice between them will have its con- 
tents equal, approximately, to half the sum of its upper and 
lower surfaces multiplied by the vertical distance apart of the 
sections. The areas may be obtained as in L. S. (74) and 
(124). This method is used to get the cubic contents of a hill 
to be cut away ; of a hollow to be filled up ; of a great reser- 
voir in a valley, either only projected, or full of water, etc. 

(154.) Sections by Oblique Planes. This method was much 
used by the old military topographers. It is picturesque, but 
not precise. The cutting-planes are parallel, and may make 
any angle with the horizon. 



SECOND SYSTEM. 



93 



BY LINE 



CHAPTEK II. 

SECOND SYSTEM. 

OF GREATEST SLOPE, 




Fig. 110 represents an oval hill by this system 



(156.) Sketching Ground by this System. This is rapid and 
effective, but not precise. In doing this, hold the book to 
correspond with your position on the ground, and always draw 
toward you. If at the top of a hill, begin by drawing lines 
from the bottom, and vice versa. The hatchings are guided 
by contour-lines lightly sketched in. 

(157.) Details of Hatchings. They must be drawn very truly 
perpendicular to the contour-lines. But if the contour-lines 
are not parallel, the hatchings must curve. When the con- 
tours are very far apart, as on nearly level ground, then pen- 
cil in intermediate ones. 

Hatchings in adjoining rows should not be continuous, but 
" break joints," to indicate the places of the contour-lines, 
which are usually pencilled in to guide the hatchings, and 
then rubbed out. The rows of hatchings must neither overlap 



94 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

nor separate, and the lines should be made slightly tremulous. 
When they are put in without contour-lines to guide them, 
take care never to let two rows run into one ; for the breaks 
between the rows represent contour-lines, and two contour- 
lines of different heights can never meet except on a vertical 
surface. 



CHAPTER III. 



THIRD SYSTEM. 



BY SHADES FROM TEBTICAL LIGHT, 



Fig. 111. 



PiiSlilii 1 
in 



! i ! 



■MM 




ill!! 



(158.) Degree of Shade. The steeper the slope is, the less 
light it receives, in the inverse ratio of its length ; i. e., in- 
versely as the secant of the angle a which 
it makes with the horizon, or directly as 
cos. a. Then the ratio of the black to the 
white is, : : 1 — cos. a : cos. a. 

In practice, the difference of shade is 
much exaggerated. 

Tables have been prepared By various 
nations, .establishing the ratio of black and 
white. 

The proper degree of shade may be 
given to the hills and hollows on the map by various means. 

(159.) Shades by Tints. Indian-ink, or sepia, is used. The 
shades are put on with proper darkness, according to a pre- 
viously-prepared " diapason of tints." The tints are made 
light for gentle slopes, and dark for steep slopes, in a constant 
ratio, a slope of 60° being quite black, one of 30° a tint 
midway between that and white, and so on. The edges at 
the top and bottom are softened off with a clean brush. This 



THIRD SYSTEM. 95 

is rapid and effective, but not very definite or precise, except 
in combination with contonr-lines. 

(160.) Shades by Contour-Lines. This is done by making 
the contour-lines more numerous ; i. e., interpolating new ones 
between those first determined. One objection to this is con- 
fusion of these lines with roads. 

(161.) Shades by Lines of Greatest Slope. The lines of steep- 
est slope, i. e., the hatchings between the contours, have their 
thickness and distance apart made proportional to the steep- 
ness of the slope, in some definite ratio. This is the most 
usual method. 

The tints may be produced by varying the thickness of the 
hatchings, or their distance apart. Both are usually combined. 

(162.) The French Method. In this the degree of inclination 
is indicated by varying the distances between the centres of 
the hatchings. The rule is : the distance hetween the centres 
of the lines shall equal y^ of an inch, plus J of the denomi- 
nator of the fraction denoting the declivity (i. e., tangent of 
the angle made oy the surface of the ground with the plane of 
reference) expressed in hundredths of an inch. 

The lines are made heavier as the slope is steeper, being 
fine for the most gentle slopes, and increasing in breadth till 
the blank space between them equals J the breadth of the 
lines. 

Only slopes of from y to -^ inclusive are represented by 
this method. 

(163.) The German, or Lehmann's Method. He uses nine 
grades for slopes from 0° to 45°, the first being white and the 
last black. For the intermediate slopes, he makes the white 
to the black in the following proportion : 

The white : the olack :: 45°— angle of slope : angle of slope. 

For example, for 30° : 

light : shade :: 45°- 30° : 30° :: 1 : 2. 

Hence, the space between the strokes is to their thickness, 
as 45° minus the angle of the slope is to the angle of the slope. 



96 LEVELLING, TOPOGKAPHY, AND HIGHER SURVEYING. 
Slopes steeper than 45° are represented by short, heavy lines, 

Fig. 112. 



II 



o c 



10° 15° 20° 25° 30' 




parallel to the contour-lines, as shown in the upper right-hand 
corner of Fig. 113 — a hill drawn by Lehmann's Method. 

Fig. 113. 







FOURTH SYSTEM. 
(164.) Another Diapason of Tints: 



97 



Slope. 


2i° 


5° 


10° 


15° 


25° 


35° 


45° 


60° 


75° 


Black. 


1 


2 


3 


4 


5 


6 


V 


8 


9 


White. 


10 


9 


8 


V 


6 


5 


4 


3 


2 



This distinguishes gentle slopes better. It makes them 
darker, and the steeper slopes lighter, and provides for slopes 
beyond 45°. To use this standard, make it on the edge of a 
strip of paper, and apply that to the map in various parts, 
and draw a few lines corresponding to the slope of those parts ; 
then fill up the intervening portions with suitable gradations. 
The angle of the slope is known from the map, since its tan- 
gent equals the vertical distance between the contours, divided 
by the horizontal distance. A scale can be made for any 
given vertical distance. 



FOUR1H SYSTEM. 



BY SHADES PRODUCED BY OBLIQUE LIGHT. 

(165.) Light is supposed to fall from the upper left-hand 
corner, as in drawing an " elevation," although the map is in 
plan. Then slopes facing the light will have a light tint,, and. 
those on the opposite side a dark tint. 

This is picturesque, but not precise. It gives apparent 
" relief" to the ground, but does not show the degree of 
steepness. 

The shades may be produced, as in the last method, by 
any means — tints, contours, or hatchings. 

By making a map with contour-lines, and shaded obliquely, 
it will be both effective and precise. 
7 



98 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



CHAPTER IY. 



CONVENTIONAL SIGNS 



(166.) Signs for Natural Surface. Scmd is represented by 
fine dots made with the point of the pen ; gravel, by coarser 
dots. Rocks are drawn in their proper places, in irregular 
angular forms, imitating their true appearance as seen from 
above. The nature of the rocks, or the geology of the country, 
may be shown by applying the proper colors, as agreed on by 
geologists, to the back of the map, so that they may be seen 
by holding it up against the light, while they will thus not 
confuse the usual details. 



(167.) Signs for Vegetation. Woods are represented by scol- 
loped circles, irregularly disposed, imitating trees seen " in 

plan," and closer or farther apart 
according to the thickness of the 
forest. It is usual to shade their 



Fig. 114. 




lower and right-hand sides, and to 



represent their shadows, as in the 
figure, though, in strictness, this is 
inconsistent with the hypothesis of 
vertical light, usually adopted for " hill-drawing." . For pine 
and similar forests, the signs may have a star-like form, as on 
the right-hand side of the figure. Trees are sometimes drawn 
"in elevation," or sideways, as usually seen. This makes 
them more easily recognized, but is in utter violation of the 
principles of mapping in horizontal projection, though it may 
be defended as a pure convention. Orchards are represented 
by trees arranged in rows. Bushes may be drawn like trees, 
but smaller. 



CONVENTIONAL SIGNS. 



Fig. 115. 






T«w vv-v 

^ vyv 









\ vv 

- vvy 
vvm 



Grass-land is drawn with irregularly scattered groups of 
short lines, as in the figure, the lines be- 
ing arranged in odd numbers, and so 
that the top of each group is convex, 
and its bottom horizontal or parallel to 
the base of the drawing. Meadows are 
sometimes represented by pairs of di- 
verging lines (as on the right of the fig- 
ure), which may be regarded as tall blades of grass Unculti- 
vated land is indicated by appropriately intermingling the 
signs for grass-land, bushes, sand, and rocks. 
Cultivated land is shown by parallel rows of 
broken and dotted lines, as in the figure, 
representing furrows. Crops are so tempo- 
rary that signs for them are unnecessary, - 
though often used. They are usually imita- B^~~-~I-^~E-^s: 

tive, as for cotton, sugar, tobacco, rice, vines, : ' "-"■* 

hops, etc. Gardens are drawn with circular and other beds 
and walks. 



Fig. 116. 



L 



(168.) Signs for Water. The /Sea-coast is represented by 
drawing a line parallel to the shore, following all its windings 



Fig. 117. 




and indentations, and as close to it as possible ; then another 
parallel line a little more distant ; then a third still more dis- 



100 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

taut,' and so on, as in Fig. 117. If these lines are drawn from 
the low-tide mark, a similar set may be drawn between that 
and the high-tide mark, and dots, for sand, be made over the 
included space. 

Bivers have each shore treated like the 
sea-shore, as in Fig. 118. 

Brooks would, be shown by only two 
lines, or one, according to their magnitude. 
Bonds may be drawn 
like sea-shores, or rep- 
resented by parallel 
horizontal lines ruled 
across them. Marshes 
and Swamps are repre- 
sented by an irregular 
intermingling of the preceding sign with 
that for grass and bushes, as shown in Fig. 119. 



Fig. 119. 





(169.) Colored Topography. The conventional signs which 
have been described, as made with the pen, require much time 
and labor. Colors are generally used by the French as sub- 
stitutes for them, and combine the advantages of great rapidity 
and effectiveness. Only three, colors (besides Indian-ink) are 
required, viz. : Gamboge (yellow), Indigo (blue), and Lake 
(scarlet), Sepia, Burnt Sienna, Yellow Ochre, Eed Lead, and 
Vermilion, are also sometimes used. The last three are diffi- 
cult to work with. To use these paints, moisten the end of a 
cake and rub it up with a drop of water, afterward diluting 
this to the proper tint, which should always be light and deli- 
cate. To cover any surface with a uniform flat tint, use a 
large camel's-hair or sable brush, keep it always moderately 
full, incline the board toward you, previously moisten the 
paper with clean water if the outline is very irregular, begin 
at the top of the surface, apply a tint across the upper part, 
and continue it downward, never letting the edge dry. This 
last is the secret of a smooth tint. It requires rapidity in 



CONVENTIONAL SIGNS. 101 

returning to the beginning of a tint to continue it, and dex- 
terity in following the outline. Marbling, or variegation, is 
produced by having a brush at each end of a stick, one for 
each color, and applying first one, and then the other, beside 
it before it dries, so that they may blend, but not mix, and 
produce an irregularly-clouded appearance. Scratched parts 
of the paper may be painted over by first applying strong 
alum-water to the place. 

The conventions for Colored Topography, adopted by the 
French military engineers, are as follows: Woods, yellow ; 
using gamboge and a very little indigo. Grass-land, green ; 
made of gamboge and indigo. Cultivated land, brown y lake, 
gamboge, and a little Indian-ink ; " burnt sienna " will an- 
swer. Adjoining fields should be slightly varied in tint. 
Sometimes furrows are indicated by strips of various colors. 
Gardens are represented by small rectangular patches of 
brighter green and brown. Uncultivated land, marbled green 
and light brown. Brush, brambles, etc., marbled green and 
yellow. Heath, furze, etc., marbled green and pink. Vine- 
yards, purple ; lake and indigo. Sands, a light brown / gam- 
boge and lake ; " yellow ochre " will do. Lakes and rivers, 
light blue, with a darker tint on their upper and left-hand 
sides. Seas, dark blue, with a little yellow added. Marshes, 
the blue of water, with spots of grass, green, the touches all 
lying horizontally. Roads, brown; between the tints for 
sand and cultivated ground, with more Indian-ink. Hills, 
greenish brown / gamboge, indigo, lake, and Indian-ink. Woods 
may be finished up by drawing the trees as in Art. (167), and 
coloring them green, with touches of gamboge toward the 
light (the upper and left-hand side), and of indigo on the op- 
posite side. 

(170.) Signs for Miscellaneous Objects. Too great a number 
of these will cause confusion. A few leading ones will be 
given : 



102 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



Signal of survey, 


/l\ 120 


Saw mill, 


€| 129 


Telegraph, 


$W 121 


Wind mill, 


@X 130 


Court house, 


fij 122 


Steam mill, 


W 131 


Post office, 


JiPl 123 


Furnace, 


4 132 


Tavern, 


mm, 124 


Woollen factory . 


# 133 


Blacksmith's shop, 




^m 125 


Cotton factory, 


# 134 


Guide board, 


1 126 


Glass works, 


A 135 


Quarry, 


X 12V 


Church, 


<?> 136 


Grist mill, 


•0 128 


Grave yard, 


—QL 137 



Stone bridge, 
Wooden bridge, 

Suspension bridge, 

Aqueduct, 

Dam, 



Fig. 138. 

An ordinary house is drawn 
in its true position and size, 
and the ridge of its roof shown, 
if the scale of the map is large 
enough. On a very small scale, 
a small shaded rectangle rep- 
resents it. If colors are used, 
buildings of masonry are tinted 
a deep crimson (with lake), and 
those of wood with Indian-ink. 
Their lower and right-hand sides 
are drawn with heavier lines. 
Fences of stone or wood, and 
hedges, may be drawn in imita- 
tion of the realities ; and, if de- 
sired, colored appropriately. 

Mines may be represented 
by the signs of the planets, 

which were anciently associated with the various metals. The 

signs here given represent respectively: 




Boat ferry, 



Rope ferry, 



Steam ferry, 



Ford for carriages, 



Ford for horses. 



Gold. 


Silver. 


Iron. 


Copper. 


Tin. 


Lead. 


Quicksilver. 


m 


E> 


6 


$ 


U 


^ 


$ 



CONVENTIONAL SIGNS. 103 

A large black circle, ©, may be used for Coal. 

Boundary -lines, of private properties, of townships, of 
counties,. and of States, may be indicated by lines formed of 
various combinations of short lines, dots, and crosses, as below. 1 



+ + + + + + + + + + + + + + + + + + + 

(171.) Scales. The scale to which a topographical map 
should be drawn, depends on several considerations. The 
principal ones are these : It should be large enough to express 
all necessary details, and yet not so large as to be unwieldy. 
The scale should be so chosen that the dimensions measured 
on the ground can be easily converted, without calculation, 
into the corresponding dimensions on the map. 

In the United States Engineer service, the following scales 
are prescribed : 

General plans of buildings, 1 inch to 10 feet (1 : 120). 

Maps of ground, with horizontal curves one foot apart, 1 inch to 50 feet (1 : 600). 

Topographical maps, one mile and a half square, 2 feet to one mile (1 : 2,640). 

Do. comprising three miles square, 1 foot to one mile (1 : 5,280). 

Do. between four and eight miles square, 6 inches to one mile (1 : 10,560). 

Do. comprising nine miles square, 4 inches to one mile (1 : 15,840). 

Maps not exceeding 24 miles square, 2 inches to one mile (1 : 31,680). 

Maps comprising 50 miles square, 1 inch to one mile (1 : 63,360). 

Maps comprising 100 miles square, ^ inch to one mile (1 : 126,720). 

Surveys of roads, canals, etc., 1 inch to 50 feet (1 : 600). 

On the admirable United States Coast Survey, all the 
scales are expressed fractionally and decimally. The surveys 
are generally platted originally on a scale of one to ten or 
twenty thousand, but in some instances the scale is larger or 
smaller. 

1 Very minute directions for the execution of the details of topographical map- 
ping, are given in Lieutenant R. S. Smith's " Topographical Drawing." Wiley, 
New York. 



104: LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

These original surveys are reduced for engraving and pub- 
lication, and, when issued, are embraced in three general 
classes : 1°, Small Harbor Charts ; 2°, Charts of Bays, Sounds ; 
and 3°, of the Coast General Charts. 

The scales of the first class vary from 1 : 10,000 to 1 : 60,000, 
according to the nature of the harbor and the different objects 
to be represented. 

Where there are many shoals, rocks, or other objects, as in 
Nantucket Harbor and Hell-Gate, or where the importance 
of the harbor makes it necessary, a larger scale of 1 : 5,000, 
1 : 10,000, and 1 : 20,000, is used. But where, from the size 
of the harbor, or its ease of access, a smaller one will 
point out every danger with sufficient exactness, the scales of 
1 : 40,000 and 1 : 60,000 are used, as in the case of New-Bed- 
ford Harbor, Cat, and Ship Island Harbor, New Haven, etc. 

The scale of the second class, in consequence of the large 
areas to be represented, is usually fixed at 1 : 80,000, as in the 
case of New- York Bay, Delaware Bay and River. Preliminary 
charts, however, are issued, of various scales from 1 : 80,000 
to 1 : 200,000. 

Of the third class, the scale is fixed at 1 : 400,000 for the 
general chart of the coast from Gay Head to Cape Henlopen, 
although considerations of the proximity and importance of 
points on the coast may change the scales of charts of other 
portions of our extended coast. 



PART V. 
UNDERGROUND OR MINING SURVEYING. 



(172.) It has three objects : 

1. To determine the directions and extent of the present 
workings of a mine. 

2. To find a point on the surface of the ground from which 
to sink a shaft, to meet a desired spot of the underground 
workings. 

3. To direct the underground workings to meet a shaft or 
any other desired point. 

It attains these objects by a combination of surveying and 
levelling: 



CHAPTER I. 



SURVEYING AND LEVELLING OLD LINES. 

(173.) First Object. To determine the direction and extent 
of the present workings of a mine. 
"We have to measure : 

1. Azimuths, or directions right and left. 

2. Lengths or distances. 

3. Heights, or distances up and down, either by perpen- 
dicular or angular levelling ; usually, the latter. 



106 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

This being done, the relative positions of all the points are 
known by their three rectangular coordinates. 

They are referred, 1st, to a vertical plane (which may be 
either north and south, or pass through the first line of the 
survey) ; 2d, to another vertical plane, perpendicular to the 
preceding one ; and 3d, to a horizontal datum-plane. 

(174.) The Old Method. In the old method of Mining Sur- 
veying, a compass * is used for determining the azimuths. One 
form of the compass used for this purpose, and called a " dial," 
is divided from 0° to 360°. In a " right-hand dial," so called, 
the numbers run as on a watch-face ; the 90° point being then 
on the east. In a " left-hand dial " they run in a contrary 
direction. In each, the zero-point goes ahead, and the 180°- 
point is at the eye. 

The bearings are taken in the usual manner, a lamp being 
the object sighted to, instead of the rod used in work on the 
surface. 

To test the accuracy of the bearing of a line taken at one 
end of it, set up the compass at the other end, or point sighted 
to, and look back to a lamp held at the first station, or point 
where the compass had been placed originally. The reading 
of the needle should now be the same as before. 

If the position of the sights had been reversed, the reading 
would be the Reverse Bearing ; a former bearing of JNf. 30° E. 
would then be S. 30° W., and so on. 

If .the back-sight does not agree with the first or forward 
sight, this latter must be taken over again. If the same dif- 
ference is again found, this shows that there is local attraction 
at one of the stations ; i. e., some influence, such as a mass of 
iron-ore, ferruginous rocks, etc., which attracts the needle, and 
makes it deviate from its usual direction. 

To discover at which station the attraction exists, set the 
compass at several intermediate points in the line which joins 

1 For a complete description of the compass, and method of using it, the va- 
riation of the magnetic needle, and methods of determining the true meridian, see 
L. S. Part III. 



SURVEYING AND LEVELLING OLD LINES. 



107 



Fig. 140. 



the two stations, and take the bearing of the line at each of 
these points. The agreement of several of these bearings, 
taken at distant points, will prove their correctness. 

When the difference occurs in a series of lines, proceed 
thus : Let C be the station FlG m 

at which the back-sight to 
B differs from the fore- 
sight from B to C. Since 
the back-sight from B to A 
is supposed to have agreed with the fore-sight from A to B, 
the local attraction must be at C, and the forward bearing 
must be corrected by the difference just found between the 
fore and back sights, adding or subtracting it, according to 
circumstances. An easy method is to draw a figure for the 
case, as in Fig. 140. In it, suppose 
the true bearing of B C, as given 
by a fore-sight from B to C, to be 
K40 o E., but that there is local 
attraction at C, so that the needle 
is drawn aside 10°, and points in 
the direction S'JST^, instead of SK 
The back-sight from C to B will 
then give a bearing of N. 50° E. ; a 
difference, or correction for the next 
fore-sight, of 10°. If the next fore- 
sight, from C to D, be K' 70° E., this 10° must be subtracted 
from it, making the true fore-sight N". 60° E. 

A general rule may also be given. When the hack-sight is 
greater than the fore-sight, as in this case, subtract the differ- 
ence from the next fore-sight, if that course and the preceding 
one have both their letters the same (as in this case, both be- 
ing !N". and E.), or both their letters different ; or add the dif- 
ference if either the first or last letters of the two courses are 
different. When the hacJc-sight is less than the fore-sight, add 
the difference in the case in which it has just been directed to 
subtract it, and subtract it where it was before directed to 
add it. 




108 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

When the compass indicates much local attraction, the 
difference between the directions of two meeting lines (or the 
" angle of deflection " of one from the other), can still be cor- 
rectly measured, by taking the difference of the bearings of 
the two lines, as observed at the same point. For, the error 
caused by the local attraction, whatever it may be, affects both 
bearings equally, inasmuch as a " bearing" is the angle which 
a line makes with the direction of the needle, and that here 
remains fixed in some one direction, no matter what, during 
the taking of the two bearings. Thus, in Fig. 140, let the true 
bearing of B C, i. e., the angle which it makes with the line 
SN, be, as before, K 40° E., and that of C D K 60° E. 
The true " angle of deflection " of these lines, or the angle 
B'C D, is therefore 20°. Now, if local attraction at C causes 
the needle to point in the direction $'W, 10° to the left of its 
proper direction, B C will bear K 50° E., and C D K 70° 
E., and the difference of these bearings, i. e., the angle of de- 
flection, will be the same as before. 

In chaining, the leader holds a lamp in the same hand 
with the end of the chain, so as to be put in line. When this 
is done, the follower drops his end of the chain, and goes on 
to find the pin, or mark made by the leader, before the leader 
leaves it. 

A gallery of a mine is thus surveyed like a road. 

To measure angles of elevation or depression of the floor 
of the gallery, a fine string or wire is stretched parallel to the 
slope, and to it a divided semicircle is attached, and the angle 
noted by a plumb-line suspended from its centre. See Fig. 71. 

(175.) The New Method. The work by the old method is 
very imperfect, owing to the variation of the magnetic needle, 
the liability of error from local attraction, and the want of 
precision in reading the angles, both horizontal and vertical. 

A transit, or theodolite, should be used. The azimuthal 
and vertical angles are taken at the same time ; the former on 
the horizontal graduated circle, and the latter on the vertical 
circle. 



SUEVEYING AND LEVELLING OLD LINES. 109 

Instead of measuring the angles which each line makes 
with the magnetic meridian, as when the compass is used, the 
angles measured are those which each line makes with the 
preceding one, or with the first line of the survey, if the method 
of traversing be adopted. 

The polar coordinates given by the transit are to be re- 
duced to the three coordinate planes, to obtain the rectangular 
coordinates. 

Yery great accuracy can be obtained by using three tri- 
pods. One would be set at the first station and sighted back 
to from the instrument placed at the second station, and a 
forward sight be then taken to the third tripod, placed at the 
third station. The instrument would then be set on this third 
tripod, a back-sight taken to the tripod remaining on the sec- 
ond station, and a fore-sight taken to the tripod brought from 
the first station to the fourth station, to which the instrument 
is next taken ; and so on. Two lamps, fitting on the tripods, 
are provided, to which the backward and forward sights are 
directed. 

Owing to the irregularity of mines, and the obstacles to 
be overcome, great difficulties exist in mining surveying. One 
is that of setting up the transit. When it cannot be set upon 
the tripod, it is often set upon sockets which are fastened to 
the wall or roof of the mine. 

(176.) The Mining Transit. In this the telescope is on one 
side, as shown in Fig. 141, and is balanced by a weight on the 
opposite side. The advantage of 

. . . Fig. 141. 

this form is, that sights may be 
taken vertically up or down, as is 
sometimes necessary in connect- 
ing the underground surveys with 
those on the surface. 

(177.) Mapping. The galleries 
of a mine on the same "level" 
may be platted in the same manner as a road or stream, etc. 




HO LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

When different "levels" are to be represented, with their 
connecting shafts, etc., " isometrical projection " has been 
used, but " military or cavalier projection " is best. 



CHAPTER II. 

LOCATING NEW LINES. 

(178.) Second Object. To determine, on the surface of the 
ground, where to sink a shaft to meet a desired point in the 
underground workings. 

To do this, repeat on the surface of the ground the survey 
made under it ; i. e., trace on it the courses and distances of 
the galleries, or their equivalents. Art. (182). 

-.The chief difficulty is to get a starting-point, and to deter- 
mine the direction of the first line. 

(179.) When the Mine is entered by an Adit, Fig. 142. Set 
the theodolite at the entrance, and get the direction of the adit, 

Fig. 142. 



and prolong it up the hill ; i. e., in the same vertical plane. 
The third adjustment is here important. See Art. (93), 

If the line has to be prolonged by setting the instrument 
farther on, the second adjustment is important. Art. (93). 



LOCATING NEW LINES. HI 

(180.) When the Mine is entered by a Shaft. Get the mag- 
netic bearing of the first underground line, at the bottom of 
the shaft, with great care. Bring up the end of the line 
through the shaft by a plumb-line, and set the compass over 
this point. Set out a line with the same bearing and length 
as the first underground line, and repeat the succeeding courses. 

When the compass cannot be set over the point, proceed 
thus : 1st. Find, by trial, a 
spot, as B (Fig. 143), which FlG - m 

is in the correct course, and ,,q 

measure off a distance equal ^-'' 

to the length of the first un- ^ 

derground course, and then 
proceed as before. 

2d. Otherwise. — Set up 
anywhere, as at A', Fig. 144, 
take the bearing and distance 

of A from A' ; run a line, corresponding with the one under- 
ground, from A' to B'. Repeat the course A' A from B' B ; 
then A B is the desired line. 



Fig. 144. 





(181.) To dispense with the Magnetic Needle. First Method. 
Let down two plumb-lines on opposite sides of the shaft, so 
that their lower ends shall be very precisely in the under- 
ground line. The plumbs may be immersed in water to pre- 
vent vibration. The plumb-lines at the top of the shaft will 
give the required line on the surface ; but its shortness is bad. 

Second Method. — Set, by repeated trials, two transits on 
opposite sides of the shaft, so that they shall at the same time 
point to one another, and each, also, to one of two points in 



112 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

the underground line. They will then give the direction of 
the line above-ground. 

Third Method. — If the telescope of the transit be eccen- 
tric, as in Fig. 141, set the instrument on a platform over the 
mouth of the shaft, so that the line of collimation of the tele- 
scope shall be in the same vertical plane with two points in 
the underground line, on opposite sides of the shaft. "When 
the instrument is so placed that, in turning the telescope, the 
intersection of the cross-hairs strikes the two points in the 
underground line, the line of sight, when directed along the 
surface, will give the required line. 

(182.) Having determined the first line, the courses of the 
underground survey may be repeated on the surface ; or the 
bearing and length of a single line be calculated, which shall 
arrive at the desired point. 

Let the zigzag line, AB, BC, CD, DZ, Fig. 145, be the 

_ ,,„ courses surveved underground, A being an adit, 

Fig. 145. J ta ' . 

i or at the bottom of a shaft, and Z the point to 

which it is desired to sink a shaft. It is required i 
to find the direction and length of the straight 
line A Z. 

When the compass is used, calculate the lati- 
tude and departure of each of the courses, A B, 
B C, etc. The algebraic sum of their latitudes 
will be equal to A X, and the algebraic sum of 
their departures will be equal to X Z. Then is 

X 7 
tan. ZAX= y~\'> *• e -> ^he algebraic sum of 

the departures divided by the algebraic sum of the latitudes is 
equal to the tangent of the bearing. The length of the line 
A Z equals the square root of the sum of the squares of A X 
and X Z ; or equals the latitude divided by the cosine of the 
bearing. 

When the transit is used, instead of referring all of the 
lines to the magnetic meridian, as in the preceding case, any 
line of the survey may now be taken as the meridian, as in 
" traversing." 



LOCATING NEW LINES. 



113 



In Fig. 146 all of the courses are referred to the first line 
of the survey. As before, a right-angled tri- 

X Z 

angle will be formed. Tan. ZAX= yr' 



Fig. 146. 



and the length of A Z = V A X* + X Z* ; or 
AX-^cos. XAZ. 

Two or more lines may be substituted 
for the single line in the two preceding 
cases; the condition being, that the alge- 
braic sums of their latitudes and of their 
departures shall be equal to those of the underground survey. 




Fig. 147. 



(183.) Third Object. To direct the workings of a mine to 
any desired point. 

This is the converse of the second object. We repeat under 
the ground the courses run above-ground ; or their equivalents, 
as in Art. (182). 

In Fig. 147, let A B, B C, CD, D Y, be the present work- 
ings of a mine, and Z the shaft to which the 
workings are to be "directed. 

Find the latitude and departure of A Z. 
Then the difference between the algebraic 
sum of the latitudes of the underground 
courses already run, and the latitude of A Z, 
is the latitude of the required course ; and 
the difference between the algebraic sum of 
the departures of the underground lines, and 
the departure of A Z, is the departure of the 
required course. 

The length of Y Z equals the square root 
of the sum of the squares of its latitude and departure. 




(184.) Problems. Most of the problems which arise in 
Mining Surveying can be solved by an application of the fa- 
miliar principles of geometry and trigonometry. 

1. Given, the angle which a vein makes with the horizon, 

8 



shaft at D will be required to 

Fig. 148. 



A 



114 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

and the place where it meets the surface, to find how deep a 

shaft at D wil 
D strike the vein : 

DC = AD.tan. DAC. 

ic 

2. Given, the depth of the shaft 

I) C, and the " dip " of the vein, to find where it crops out : 

AD = DC.cot.DAC. 

3. Given, the depth of a shaft when the vein " crops out," 
and the " dip " of the vein, to find the distance from the bot- 
tom of the shaft to the vein : 

BC = AB.cot. ACB. 

If the ground makes an angle with the horizon, then the 
problems involve oblique-angled triangles instead of right- 
angled triangles, as in the preceding cases. Their solution, 
however, is quite as simple. 

In the more difficult problems, the measurement of lines is 
required, one or both ends of which are inaccessible. For a 
full investigation of this subject, see " Gillespie's Land Sur- 
veying,' 



PART VI. 

TEE SEXTANT, AND OTHER REFLECTING 
INSTRUMENTS. 



CHAPTEK I. 



THE INSTRUMENTS. 

(185.) Principle. The angle subtended at the eye by lines 
passing from it to two distant objects, may be measured by so 
arranging two mirrors that one object is looked at directly, 
and the other object is seen by its image, reflected from one 
mirror to the second, and from the second mirror to the eye. 
If the first mirror be moved so that the doubly-reflected image 
of the second object be made to cover or coincide with the 
object seen directly, then is the desired angle equal to twice 
the angle which the mirrors make with each other. 

Proof. — Let two mirrors be parallel. Then a ray of light, 
striking one of them, reflected to the other, and reflected again 
from that, would pass off in a direction parallel to its first 
direction. 




Let a equal the angle between the incident ray and the 



116 LEVELLING, TOPOGKAPHY, AND HIGHER SURVEYING. 

first mirror. ]STow, suppose the first mirror to be turned n°. 
The incident ray now makes an angle with this mirror n° 
greater than before ; it will therefore pass off, making an an- 
gle with the mirror n° greater than before. But the mirror 
itself now makes an angle of n° with its former direction ; 
therefore, the ray will pass off at an angle of a° + 2n° with 
the former surface of the mirror, and in a direction differing 
2n° from its former direction, and the direction of the ray re- 
flected from the second mirror will therefore differ 2n° from 
its former direction. 




If, now, an eye, placed at E, sees an object in the second 
mirror, in the direction E H, which has been reflected from 
two mirrors, then the line E H makes an angle with the true 
direction of the line equal to twice the angle which the mir- 



THE INSTRUMENTS. 



117 



rors make with one another. If the eye also sees an object, 
directly in the line E H, which apparently coincides with the 
reflected image of the first object, then is the angle, subtended 
at the eye by the lines passing to it from the two objects, 
equal to twice the angle which the two mirrors make with one 
another. 

(186.) Description of the Sextant, Fig. 150. The frame is 
usually of brass, constructed so as to combine strength with 
lightness. The handle, H, by which it is held, is of wood. 
AB is a graduated arc ; C D, the index-arm, is movable about 
a pivot in the centre of the graduated arc. M is a glass, which 
may be moved over the vernier to aid in reading* it. The in- 
dex-glass, I, is a small mirror, attached to the index-arm, so 
as to be perpendicular to the plane of the graduated arc. The 

Fig. 151. 




horizon-glass, H, is attached perpendicularly to the plane of 
the instrument, and parallel to the index-glass when the index 
is at zero. The lower half of this glass is silvered, to make it 
a reflector, and the upper half is transparent. J 1 E is the tel- 



118 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

escope ; S S are sets of colored glasses, used to moderate the 
light of the sun, when that body is observed. 

The sextant has an arc of one-sixth of a circle, and meas- 
ures angles up to 120°, the divisions of the graduated arc 
being numbered with twice their real value, so that the true 
desired angle, subtended by the two objects, is read off at 
once. The arc is usually graduated to 10' and subdivided by 
a vernier to 10". 

(187.) The box or pocket sextant, shown in Fig. 151, has 
the same glasses as the larger sextant, enclosed in a circular 
box, about three inches in diameter. The lower part, which 
answers for a handle when in use, screws off and is used for a 
cover, making the instrument only half as deep as it appears 
in the figure. 

The octant has an arc of one-eighth of a circumference, 
and measures angles to 90°. 

(188.) The Reflecting Circle. This is an instrument con- 
structed on the same principle, and used for the same pur- 
poses, as the sextant. In it the graduated arc extends to the 
whole circumference, and more than one vernier may be used 
by producing the index-arm to meet the circumference in one 
or two more points. 

(189.) Adjustments of the Sextant. 1. To make the index- 
glass perpendicular to the plane of the arc : 

Bring the index near the centre of the arc, and place the 
eye near the index-glass, and nearly in the plane of the arc. 
See if the part of the arc reflected in the mirror appears to be 
a continuation of the part seen directly. If so, the glass is 
perpendicular to the plane of the arc. If not, adjust it by 
the screws behind it. 

2. To make the horizon-glass perpendicular to the plane of 
the arc : 

The index-glass having been adjusted, sight to some well- 
defined object, as a star, and if, in moving the index-arm, 
one image seems to separate from or overlap the other, then 



THE INSTRUMENTS. 119 

the horizon-glass is not perpendicular to the plane of the arc. 
It must be made so by the screws attached to it. 

Another method of testing the perpendicularity of the 
horizon-glass is as follows: Hold the instrument vertically, 
and bring the direct and reflected images of a smooth portion 
of the distant horizon into coincidence. Then turn the instru- 
ment until it makes an angle with the vertical. If the two 
images still coincide, the glasses are parallel; and, as the 
index-glass has been made perpendicular to the plane of the 
arc, the horizon-glass is in adjustment. 

3. To make the line of collimation of the telescope parallel 
to the plane of the arc : 

The line of collimation of the telescope is an imaginary 
line, passing through the optical centre of the object-lens, and 
a point midway between the two parallel wires. These wires 
are made parallel to the plane of the sextant by revolving the 
tube in which they are placed. 

To see whether the line of collimation of the telescope is 
in adjustment, bring the images of two objects, such as the 
sun and moon, into contact at the wire nearest the instrument, 
and then, by moving the instrument, bring them to the other 
wire. If the contact remains perfect, the line of collimation 
is parallel to the plane of the arc ; if it does not, the adjust- 
ment must be made by the screws in the collar of the tele- 
scope. 

4. To see if the two mirrors are parallel when the index is 
at zero : 

Bring the direct and reflected images of a star into coin- 
cidence. If the index is at zero, then no correction is neces- 
sary ; if not, the reading is the " index-error", and is positive 
or negative, according as the index is to the right or left of 
the zero. 

The "index-error" may be rectified by moving the hori- 
zon-glass until the images do coincide when the index is at 
zero, but it is usually merely noted, and used as a correction, 
being added to each reading if the error is positive, or sub- 
tracted from each reading if the error is negative.. 



120 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(190.) How to observe. Hold the instrument so that its 
plane is in the plane of the two objects to be observed, and 
hold it loosely. Look through the eye-hole, or plain tube, or 
telescope, at the left hand or lower object, by direct vision, 
through the unsilvered part of the horizon-glass. Then move 
the index-arm till the other object is seen in the silvered part 
of the horizon-glass, and the two are brought to apparently 
coincide. Then the reading of the vernier is the angle desired. 

If one object be brighter than the other, look at the former 
by reflection. If the brighter objects be to the left or below, 
hold the instrument upside down. 

If the angular distance of the object be more than the 
range of the sextant (about 120°), observe from one of them to 
some intermediate object, and thence to the other. 

A good rest for a sextant is an ordinary telescope-clamp, 
through which is passed a stick, one end of which is fitted 
into a hole made in the sextant-handle, and the other end of 
which is weighted for a counterpoise. 

Fig. 153. 




(191.) Parallax of the Sextant. The angle observed with 
the sextant is that made by two lines : one, B I, passing from 
the reflected object to the index-glass, and which is thence 
reflected to the horizon-glass, and thence to the eye ; and the 
other, HE, passing from the object directly to the eye; i. e., 
the angle which B I produced makes with H E. But the eye 
may be at E' on either side of E. Then we require the angle 



THE PRACTICE. 121 

which. B E' makes with H E. These angles are the same only 
when the eye is in the same line with B I prodnced ; i. e., 
when E' coincided with E. In all other cases, the observed 
angle differs from the desired angle by the small angle E B E', 
which is called the parallax of the instrument. 

It is the angle which would be subtended at the reflected 
object B by the distance E E'. It is usually very small for dis- 
tant objects. Thus, at a mile's distance, 1 inch subtends an 
angle of only 3 seconds, and of 3 minutes at 100 feet distance. 

To escape it, if one object be distant and the other near, 
view the former by reflection. If both be near, find some dis- 
tant point in line with one of them, and view this new point 
by reflection, and the other near one directly. 



CHAPTER II. 



THE PRACTICE 



(192.) To set out Perpendiculars. Set the index at 90°. 
Hold the instrument over the given point by a plumb-line, 
and look along the line by direct vision. Send a rod in about 
the desired direction, and when it is seen by reflection to coin- 
cide with the point on the line looked at directly, it will be in 
a line perpendicular to the given line at the desired point. 

'Conversely, to find where a perpendicular from a given 
point would strike a line : 

Set the index at 90°, and walk along the line, looking 
directly at a point on it, until the given point is seen by re- 
flection to coincide with the point on the line. A plumb-line 
let fall from the eye will give the desired point. 

(193.) The Optical Square, Fig. 153. This is a box contain- 
ing two mirrors, fixed at an angle of 45° to each other, and 



122 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

» 

therefore giving an angle of 90°, as does the sextant with its 
glasses fixed at that angle. It is used to set out perpendiculars. 




(194.) To measure a Line, one End being inaccessible. 

A B be the required line, and B the inaccessible point. 



Let 



Fig. 154. 




At A set off a perpendicular, A C, by Art. (192). Then 
set the index at 45°, and walk backward from A in the line 
C A prolonged, looking by direct vision at C, until you arrive 
at some point, D, from which B is seen by reflection to coincide 
with d Then isAD^AB. 

If more convenient, after setting off the right angle, set 
the index at 63° 26', and then proceed as before. The objects 
will be seen to coincide when at some point D'. Then 
AD'^JAB. If the index be set at 71° 34' ', then the meas- 
ured distance will be -JAB, and so on. 



THE PRACTICE. 



123 



If the index be set at the complements of the above angles, 
the distance measured will be, in the first case, twice, and in 
the second case three times the desired one. 

When the distance AD cannot be measured, as in Fig. 
155, fix D as before. Set the index at 26° 34/, and go along 

Fig. 155. 




the line to E, where the objects are seen to coincide with each 
other; then is AE twice AB, and hence ED = AE. 

Fig. 156 





124: LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(195.) Otherwise. At A set off an angle, asCAD (AD 
being a prolongation of AB). Then walk along the line A C 
with the index set to half that angle, looking at A directly, 
and B by reflection, till yon come to some point, 0, at which 
they coincide. Then is C A — A B. 

(196.) To measure a Line when both Ends are inaccessible. 

Let AB be the required line. At any point, C, measure 

Fig. 157. 




the angle A C B. Set the sextant to half that angle, and. walk 
back in the line B C prolonged till at some point, D, A and B 
are seen to coincide, as in last problem ; thus making A C = 
C D. Do the same on A C produced to some point, E. Then 
isDE = AB. 

(197.) All the methods for overcoming obstacles to measure- 
ment, determining inaccessible distances, etc. (L. S. Part TIL), 
with the transit or theodolite, can be executed with the sextant. 

(198.) To measure Heights. Measure the vertical angle be- 
tween the top of the object and a mark at the height of the 
eye, as with a theodolite or transit, and then calculate the 
height as in Part II., Art. (98). 

Otherwise. Set the index at 45°, and walk backward till 
the mark and the top of the object are brought to coincide. 
Then the horizontal distance equals the height. 



THE PKACTICE. 



125 



So, too, if the index is set at 63° 26', the height equals 
twice the distance, and so on. The ground is supposed to be 
level. 



Fig. 158. 




When the Base is inaccessible : Make C = 45°, and D = 
26° 34'. Then CD = AB. So, too, if C = 26° 34', and 
D = 18° 26 / . 

This may be used when a river flows along the base of a 
hill whose height is desired, or in any other like circumstance. 

(199,) To observe Altitudes in an artificial Horizon, In this 

Fig. 159 




method we measure the angle subtended at the eye between 
the object and its image reflected from an artificial horizon of 



126 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

mercury, molasses, oil, or water* The image of the object in 
the mercury is looked at directly, and the object itself is 
viewed by reflection. The object observed is supposed to be 
so distant that the rays from it, which strike respectively the 
index-glass and the artificial horizon, are parallel ; i. e., S and 
S', Fig. 159, are the same point. 

Then will the observed angle . S E S" be double the re- 
quired angle SEH. 

Demonstration, 
a = a', a' = a", and a!' = a'". Hence a'" = a. 
SE S" = a + a'" = 2a = 2 S E H. 

(200.) "When the sun is the object observed, to determine' 
whether it is his upper or lower limb whose altitude has been 
observed, proceed thus : 

Having brought two limbs to touch, push the index-arm 
from you. If one image passes over the other, so that the 
other two limbs come together, then you had the lower limb 
at first. If they separate, you had the upper limb. 

In the forenoon, with an inverting telescope, the lower 
limbs are parting, and the upper limbs are approaching ; and 
vice versa in the afternoon. 



Fig. 160. 




(201.) To observe very small altitudes and depressions with 
the artificial horizon : 



THE PKACTICE. 



127 



Stretch a string over the artificial horizon. Place your 
head so that you see the string cover its image in the mercury. 
Then the eye and string determine a vertical plane. 

Then observe, looking at the string by direct vision, and 
seeing the object by reflection, and you have the angle SEN, 
in Fig. 160, the supplement of the zenith distance. 

Otherwise. Fix behind the horizon-glass a piece of white 
paper with a small hole in it, and with a black line on it per- 
pendicular to the plane of the arc. 

Then look into the mercury, so as to see in it the image 
of the line. Your line of sight is then vertical, and the angle 
to the object seen by reflection is measured as before. 

(202.) To measure Slopes with the Sextant and Artificial 
Horizon. Let A B be the surface of the ground, and A F a 

Fig. 161. 




horizontal line. Mark two points equally distant from the 
eye. Measure, by the preceding method, the angles j3 and j3', 
which C A and C B make with the vertical CD. Then will 
half the difference of these angles equal the angle which the 
slope makes with the horizon. 

• Demonstration. Continue the vertical line C D to meet 
the horizontal line in F, and draw C E perpendicular to A B. 
Then the triangles CDE and A D F are similar, being right- 
angled and having the acute angles at D equal. Consequently, 
the angle D C E = D A F, which is the angle of the slope 
with the horizon. But D C E = \ (0'— 0), hence J ($'- P) = 
the angle which the slope of the ground makes with the 
horizon. 



128 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

If the points A and B be not equally distant from C, but 
yet far apart, this method will still give a very near approxi- 
mation, the error, which is additive, being -J (of — a). 

Demonstration. 

DCE = i3 / + a , -90°, 

DCE = -i3-a+ 90°, 
2~DCE = j3'-j3 + a'-a, 

(203.) Oblique Angles. When the plane of two objects, 
observed by the sextant, is very oblique to the horizon, the 
observed angle will differ much from the horizontal angle 
which is its horizontal projection, and which is the angle 
needed for platting. The projected angle may be larger or 
smaller than the observed angle. 

This difficulty may be obviated in various ways : 

1. Observe the angular distance of each object from some 
third object, very far to the right or left of both. The differ- 
ence of these angles will be nearly equal the desired angle. 

2. Note, if possible, some point above or below one of the 
objects, and on the same level with the other, and observe to 
it and the other object. 

3. Suspend two plumb-lines, and place the eye so that 
these lines cover the two objects. Then observe the horizon- 
tal angle between the plumb-lines. 

4. For perfect precision, observe the oblique angle itself, 
and also the angle of elevation or depression of each of the 
objects. With these data the oblique angle can be reduced to 
its horizontal projection, either by descriptive geometry or 
more precisely by calculation, thus : 

Let AH B be the observed angle, and A! H B' the required 
horizontal angle. 

Conceive a vertical II Z, and a spherical surface, of which 
H, the vertex of the angle, is the centre. Then will the ver- 



THE PRACTICE. 



129 



tical planes, AHA'andBH B', and the oblique plane AHB, 
cut this sphere in arcs of great circles, Z A", Z B", and A" B", 



Fig. 162. 



-3 




thus forming a spherical triangle, A" Z B", in which A"B" == h 
measures the observed angle ; Z A" = Z measures the zenith 
distance of the point A ; and Z B" = Z' measures the zenith 
distance of the point B. 

These zenith distances are observed directly, or given by 
the observed angles of elevation or depression. Then we have 
the three sides of the triangle to find the angle B = A'HB'. 

Calling P the half sum of the three sides we have : 



Bin. Z . sin. Z 

An approximate correction, when the zenith distances do 
not differ from 90° by more than 2° or 3°, is this : 

^90° _ ^^y tang. ih. sin. 1"- i^-^j cot. i h . sin. 1". 

The quantities in the parentheses are to be taken in sec- 
onds. 

The answer is in seconds, and additive. 

(204.) The advantages of the sextant over the theodolite 
are these : 

1 See Jackson's Trigonometry, page 65, Fifth Case. 



130 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

1. It does not require a fixed support, but can be used 
while the observer is on horseback, or on a surface in motion, 
as at sea. 

2. It can take simultaneous observations on two moving 
bodies, as the mo(to and a star. 

It can also do all that the theodolite can. Its only defect 
is in observing oblique angles in some cases. By these prop- 
erties it determines distances, heights, time, latitude, longi- 
tude, and true meridian, and thus is a portable observatory. 



PAET VII. 

MARITIME OR HYDROGRAPIIICAL 
SURVEYING. 



(205.) The object of this is to fix the positions of the deep 
and shallow points in harbors, rivers, etc., and thus to discover 
and record the shoals, rocks, channels, and other important 
features of the locality. 

The relative positions of prominent points on the shore 
are very precisely determined by " Trigonometrical Survey- 
ing," Part VIII. These form the basis of operations, and 
afford the means of correcting the results obtained by the less 
accurate methods employed for filling in the details. 



CHAPTEE I. 



THE SHORE. LINE 



(206.) The High-water Line, The principal points on the 
high-water line are determined by triangulating, Art. (233). 
The sections between these points are surveyed with the com- 
pass and chain ; by running a series of straight lines so as to 
follow, approximately, the shore line, and taking offsets from 



132 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

the straight lines of the survey to the bends in the shore line. 
The straight lines can be more accurately determined by 
" traversing " with the transit. Art. (94). 

(207.) The Low Water-Line. In " tidal- waters " this is more 
difficult, because low and bare for only a short time. The 
survey is best made with the sextant, observing from promi- 
nent points to three signals, by the trilinear method — Art. 
(213)— and sketching by the eye bends of the shore between 
the stations observed from. 

There should be one to observe and one to record. Let 1 
and 2, Fig. 163, be two points on the low-water line, whose 
position it is desired to determine. The observations taken 
will be as follows : 

(1:) A and B . . . 18° 



Fig. 163. 



c *> B and C . . . 20 



X 



\ 



(2.) B and C . . . 15° 
C and D . . . 45° 



.* ** When the shore is inaccessible* a 

/ 9 ' 

base line must be measured on the 
water, and points on the shore fixed by angles from its ends, 
as in Art. (232). 

(208.) Measuring the Base. 1. By sound. Sound travels 
at the rate of 1,090 feet per second, with the temperature at 
30° Fahr. For higher or lower temperatures, add or subtract 
1^- foot for each degree. If the wind blows with or against 
the movement of the sound, its velocity must be added or 
subtracted. If it blows obliquely, the correction will be its 
velocity multiplied by the cosine of the angle which the direc- 
tion of the wind makes with the direction of the sound. 

2. By measuring with the sextant the angular height of 
the mast of a vessel, then we have : 

Distance = height of mast -f- tan. of the angle. 



SOUNDINGS. 



133 



3. By astronomical observations at points 50 miles apart, 
more or less, determining their latitudes and longitudes, and 
hence knowing their distance and bearings. A vessel may be 




anchored at various points between A' and B', and thus new 
base lines be formed, from the ends of which to triangulate to 
points on the shore. 



CHAPTER II. 



SOUNDINGS. 



(209.) In a river or narrow water, the soundings may be 
taken in zigzag lines, from shore to shore, at equal intervals 
of time, as in Fig. 165. 



Fig. 165. 




134: LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(210.) On a Sea-coast. The position of the boat in the 
water, when the soundings are taken, must be determined at 
regular intervals. 



This is done in various wavs. 



(211.) From the Shore. By observing with a compass or 
transit to the boat from stations on the shore, at a given signal 
or fixed time. 

Fig. 166. 




Its place is then fixed as in Art. (232). Two observers 
are necessary. Three are better, as the third checks the other 
two. This is accurate in theory, but not in practice, simulta- 
neous observations being impracticable, and confusion fre- 
quent. Also, more men are required. 

(212.) From the Boat with a Compass. Establish signals 
along the shore, Art. (240), distinguish them by colors, or by 
the number of cross-pieces on the staff, thus : + ± + , and 
observe to them from the boat with a prismatic compass (L. S. 
232), or Burnier's compass, Art. (96). The place of the boat 
is then determined, and may be fixed on the map hj drawing, 
from the two known points, lines having the opposite bearings, 
and their intersection will be the required point. This is 
rapid and easy, but not precise. 

(213.) From the Boat with the Sextant. This is the trilinear 
method, and is the best. Two observers, or two sextants with 
one observer, are necessary. 



SOUNDINGS. 



135 



Fig. 16T. 



(214.) Teilinear Surveying is founded on the method of 
determining the position of a point 
by measuring the angles between three 
lines conceived to pass from the re- 
quired point to three known points. 
Thus, in the figure, the point P is de- 
termined by the angles APB and 
BPC, the points A, B, and 0, be- 
ing known. To fix the place of the 
point from these data is known as the 
"problem of the three points." It 
will be here solved' geometrically, instrumentally, and analyt- 
ically. 

(215.) Geometrical Solution. Let A, B, and C, be the known 
objects observed from S, the angles A S B and B S C being 
there measured. To iix this point, S, on the plat containing 




Fig. 168. 




A, B, and C, draw lines from A and B, making angles with 
AB, each equal to 90°— A SB. The intersection of these 
lines at O will be the centre of a circle passing through A and 

B, in the circumference of which the point S will be situated. 1 

1 For, the arc A B measures the angle AOBat the centre, which angle = 180° 
— 2 (90° — A S B) = 2 A S B. Therefore, any angle inscribed in the circumfer- 
ence and measured by the same arc is equal to A S B. 



136 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Describe this circle. Also, draw lines from B and 0, making 
angles with B C, each equal to 90° — B S C. Their intersec- 
tion, 0', will be the centre of a circle passing through B and 
C The point S will lie somewhere in its circumference, and, 
therefore, in its intersection with the former circumference. 
The point is thus determined : 

In the figure the observed angles, A S B and B S C, 
are supposed to have been respectively 40° and 60°. The 
angles set off are therefore 50° and 30°. The central angles 
are consequently 80° and 120°, twice the observed an- 
gles. 

The dotted lines refer to the checks explained in the latter 
part of this article. 

When one of the angles is obtuse, set off its difference from 
90° on the opposite side of the line joining the two objects to 
that on which the point of observation lies. 

When the angle AB C is equal to the supplement of the 
sum of the observed angles, the position of the point will be 
indeterminate, for the two centres obtained will coincide, and 
the circle described from this common centre will pass through 
the three points, and any point of the circumference will fulfil 
the conditions of the problem. 

A third angle, between one of the three points and a fourth 
point, should always be observed, if possible, and used like 
the others, to serve as a check. 

Many tests of the correctness of the position of the point 
determined may be employed. The simplest one is, that the 
centres of the circles, O and O', should lie in the perpendicu- 
lars drawn through the middle points of the lines A B and B C. 

Another is, that the line B S should be bisected perpendic- 
ularly by the line OO'. 

A third check is obtained by drawing at A and O perpen- 
diculars to AB and CB, and producing them to meet BO 
and B O', produced in D and E. The line D E should pass 
through S ; for, the angles B S D and B S E being right angles, 
the lines D S and S E form one straight line. 

The figure shows these three cheeks by its' dotted lines. 



SOUNDINGS. 137 

(216.) Instrumental Solution. The preceding process is te- 
dious where many stations are to be determined. They can 
be more readily found by an instrument called a Station- 
pointer, or Chorograph. It consists of three arms, or straight- 
edges, turning about a common centre, and capable of being 
set so as to make with each other any angles desired. This is 
effected by means of graduated arcs carried on their ends, or 
by taking off with their points (as with a pair of dividers) the 
proper distance from a scale of chords constructed to a radius 
of their length. Being thus set so as to make the two observed 
angles, the instrument is laid on a map containing the three 
given points, and is turned about till the three edges pass 
through these points. Then their centre is at the place of the 
station, for the three points there subtend on the paper the 
angles observed in the field. 

A simple and useful substitute is a piece of 'transparent 
paper, or ground glass, on which three lines may be drawn at 
the proper angles and moved about on the paper as before. 

(217.) Analytical Solution. The distances of the required 
point from each of the known points may be obtained analyt- 
ically. Let AB = o;BC = a; ABC = B; ASB = S; 
B S C = S'. Also, make T = 360° - S - S' - B. Let 
BAS = TJ; BCS = V. Then we shall have : 

A , TT ^ / c. sin. S' A 

Cot. TJ = cot. T ( -. — s — m + 11. 

\a . sin. 8 . cos. T / 

Y = T-U. 

g . sin. TJ a. sin. Y 



S B = '. * q ; or, = 



sin. S ' ' sin. S' ' 

g . sin. A B ' S n a . sin. CBS 

sin. 8 sin. b 

Attention must be given to the algebraic signs of the trig- 
onometrical functions. 



138 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Example. A S B = 33° 45'; B S C = 22° 30''; AB = 600 
feet; BC = 400 feet; AC =800 feet. Kequired the dis- 
tances and directions of the point S from each of the stations. 

In the triangle ABC, the three sides being known, the 
angle AB C is found to be 104° 28 / 39". The formula then 
gives the angle B A S = U = 105° 8' 10" ; whence B C S is 
found to be 94° 8' 11" ; and S B = 1042.51 ; S A = 710.193 ; 
and S C = 934.291. 

(218.) Between Stations. Positions of the boat are thus ob- 
served only at considerable distances apart, and the boat is 
rowed from one of these- points to a second one, and soundings 
taken at regular intervals of time between them. 

Fig. J 69. 




The distance apart of the soundings depends on the regu- 
larity of the bottom, the depth of the water, and the object of 
the survey. Care should be taken to leave no spot unex- 
plored. 

For great accuracy, anchor at some point, and determine 
its place as above, and then proceed to another point, paying 
out a line, fastened to the anchor, and sounding at regular 
distances. Cast anchor at the second point, go back to the 
first, take up the anchor, go on to the second, and then pro- 
ceed as before. 

The soundings, or depth of the water, are made with rods, 
chains, or lines, according to the depth and the precision re- 
quired. 



SOUNDINGS. 139 

(219.) The sounding-line should be strong and pliable. 
The lead fastened to its extremity should be shaped like the 
frustum of a cone. The size of the line and the weight of the 
lead will depend upon the depth of the water. The hand 
lead-line is limited to twenty fathoms and is marked thus : at 
three fathoms a piece of leather ; at five, a white rag ; at 
seven, a red rag ; at ten, a piece of leather with a round hole 
in it ; at thirteen, a blue rag ; at fifteen, a white rag ; at sev- 
enteen, a red rag ; at twenty, a piece of cord with two knots. 
These divisions are called marks. The other divisions called 
deeps are: at half a fathom a piece of leather with three 
points ; at one fathom a piece of leather with one point ; 
at two fathoms a piece of leather with two points ; at four, 
six, eight, eleven, fourteen, sixteen, and eighteen fathoms a 
piece of cord with a knot in it ; at nine and twelve, 
a piece of cord with two knots in it. The deep-se& FlG - ™- 
lead-line is similarly marked up to twenty fathoms. 
Each additional ten fathoms is indicated by a cord with 
an additional knot, and half-way between these a 
piece of leather marks the five fathoms. 

The length of the line should be frequently tested. 

The character of the bottom is determined by pla- 
cing tallow into a hollow in the base of the lead, which 
adheres to the material at the bottom. A barbed pike 
is sometimes attached to the base of the lead for this purpose. 1 
Fig. 170. 

1 For description of sounding apparatus, see U. S. C. S. Report, 1860 ; for 
deep-sea soundings, Reports of 1854, 1858, and 1859. 



140 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



CHAPTER III. 

TID E-W ATEES. 

(220.) The soundings taken must all be reduced to mean 
low spring-tides. 

The tides are semidiurnal oscillations of the ocean, caused 
by the combined attractions of the moon and sun, especially 
the former. The tide is not a current, but a broad, flat wave. 
The lunar one follows the moon, about 30° east of it. The 
solar one follows the sun. In the open sea, it is high water 
about 30° east of the moon. To the east of this the tide is 
ebbing ; west of this it is rising ; 90° distant it is low tide. 

The tides, being caused by the attraction of the sun and 
moon, are greatest when they act together ; i. e., at new and 
full moon, or a day or two after. These are spring-tides. 

They are least when the sun and moon act perpendicularly 
to each other. These are neap-tides. The spring-tides are the 
highest and lowest tides. The moon makes a tidal wave which 
recurs in 12 hours 24 minutes, while the sun's recurs in 12 
hours. The coincidence of these waves produces spring-tides, 
and their partial cancellation neap-tides. 

If the tide- wave met with no obstruction, the highest tides 
would be in those latitudes over which the sun and moon pass 
vertically ; but the height of the tide is most affected by local 
causes, such as the shoaling of the water, formation of the 
shore,. and the position and character of the channels. For 
example, the mean height of the tide at Cape Florida is 1.5 
feet, while in the Bay of Fundy it rises over 40 feet. 

The height of the tide is also affected by the wind and the 
state of the atmosphere. A wind in the direction in which 
the tide is moving, and a low barometer, will increase the 
height, and vice versa. 

(221.) The tide caused by the upper transit ot the moon is 
called the superior tide, and that caused by the lower transit 



TIDE-WATERS. 



141 



the inferior tide. "When the moon is north of the equator, 
the superior tide will be higher than the inferior tide. This 
difference in the height of the tides is called the diurnal 
inequality. On the Atlantic coast, the successive high waters 
and successive low waters are nearly at equal heights above 
and below the mean, with intervals of 12 hours and 24 
minutes. On the Pacific coast, the successive high and low 
tides may differ several feet in height, and several hours in 
intervals, as is shown in Fig. 171 ; a and c are successive high 
tides, and o and d low tides. 



Fig. 171. 




The lengthening or shortening of the interval between 
two high tides is called the priming and lagging of the tide. 

(222.) The heights of mountains and other points on the 
surface of the earth are often referred to the level of the sea 
as a datum plane. The mean level of the sea is the mean be- 
tween the mean of two successive high tides and the mean of 
the intermediate low tides. This is constant, while high and 
low tides vary. 

(223.) The tidal current in channels is due to the change 
of level, caused by the tidal wave. The rising of the tide 
which causes the flowing in of the current is called flood-tide ; 
and the falling of the tide, which causes the flowing out of 
the current, is called ebb-tide. The stand is the period during 
which the height of the tide remains stationary. Slack water 
is the interval of time during which there is no current. 
Where a narrow inlet connects a large inland basin with the 



142 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

sea, care should be taken not to confound the time of high 
water or low water with the time of the turn of the tide. The 
flood-tide may run for hours after the time of high water, and 
so, too, the stream of the ebb-tide after low water. 

(224.) " Establishment of the Fort." Owing to the ob- 
structions which the tidal wave meets with from the formation 
of the sea-bed as it approaches the shore, and the character 
and direction of the channels, the time of high water will 
differ for different ports in the same vicinity. In order that 
navigators, entering a port, may be able to find the time of 
high water, a standard tide-time is established, i. e., the num- 
ber of hours at which high water occurs after the moon's 
transit over the meridian. This is called the " Establishment 
of the Port." This time varies with the age of the moon. 
"When observed on the days of full or change, it is the " Vul- 
gar Establishment of the Port." The " Corrected Establish- 
ment of the Port " is the mean of the intervals between the 
times of the transit of the moon and the times of high tide 
for half a month. This is used for finding the time of high 
water on any given day, and tables are constructed, from 
observations at the principal posts, for finding the correction 
for semi-monthly inequality. 

(225.) Tide-Gauges. Tidal observations consist in recording 
the heights of the water at stated times. In order to deter- 
mine this, tide-gauges are necessary. The simplest form is a 
stick of timber, graduated to feet and inches, or tenths, and 
either set up in the w T ater, or fastened to the face of a dock, or 
pier, so that the rise of the tide may be noted upon it. The 
zero-point of each gauge is taken at or below the lowest tide, 
and is referred to a permanent "bench-mark" on the shore. 
On account of the difficulty of sustaining a timber of consider- 
able height against the force of the wind and waves, several, 
successive gauges are sometimes used — the bottom mark on 
each gauge higher up being on a level with the top line of the 
next lower. Such an arrangement is required on gentle slopes. 

On the sea-coast, where the waves make the reading of the 



TIDE-WATERS. 



143 



staff difficult, the staff may be attached to a float, enclosed in 
an upright tube, pierced with holes. The holes in the tube 
should be of such a size as to allow the water to find the mean 
height inside, and yet reduce the oscillations to very small 
limits. Permanent tide-gauges should be self-registering. For 
a description of a self-registering tide-gauge, see TJ. S. C. S. 
Eeport, 1853. 

(226.) Tide-Tables. Observations of tides may be recorded 
graphically, as on page 143,* or in the tabular form, given on 
page 144.* In the table, on page 143,* the horizontal line is 
divided into months, days, and half-days. The hours and 
minutes are noted at the feet of the vertical ordinates, in 
black for high water, and red for low water. The lines of the 
ordinates are black for high water, and red for low water. 
The heights of the tides are noted in feet and tenths at the 
side of the ordinates, and the weather at their summits. An- 
other graphical method is given in Fig. 171. 

The tabular form, used on the United States Coast Survey, 
is given on page 144.* In the column of remarks, the posi- 
tion of the gauge should be accurately described, and the 
position and height of the " bench-mark," to which the zero 
of the gauge is referred, so that the gauge may be replaced if 
disturbed. 

Table of records of tidal observations at some important 
points : 





Interval 'between time 














of moon's transit 


RISE A 


MEAN 


DURATION OF 




(southing) and time 












STATION. 


of high water. 














Difference be- 
















Mean. 


tween great- 
est and least 


Mean. 


Spring. 


Neap. 


Flood. 


Ebb. 


Stand. 




















H. M. 


H. M. 


Feet. 


Feet. 


Feet. 


H. M. 


H. M. 


H. M. 


Portland, 


11 25 


44 


8.8 


10.0 


7.6 


6 14 


6 12 


20 


Boston, 


'11 22 


44 


10.1 


13.1 


7.4 


6 16 


6 18 


09 


New York, 


8 13 


• 46 


4.3 


5.4 


3.4 


6 00 


6 25 


28 


Charlestown, 


7 13 


36 


5.3 


6.3 


4.6 


6 36 


6 09 


33 


Key West, 


9 22 


1 12 


1.4 


2.3 


0.7 


6 59 


5 25 


12 


San Francisco, 


12 03 


1 22 


3.9 


5.0 


2.9 


6 30 


5 52 


30 



144 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(227.) In rivers, a number of tide-gauges are necessary, at 
moderate distances apart, especially at the bends, because the 
tidal lines of high and low water are not parallel to one 
another. 

The soundings are to be reduced by the nearest gauge, or 
by the mean of the two between which they may be taken. 

(228.) Beacons and Buoys. Beacons are permanent objects, 
such as piles of stones with signals on them, usually on shoals 
and dangerous rocks. 

Buoys are floating objects, such as barrels, or hollow iron 
spheres or cylinders, anchored by a chain, and variously 
painted, to indicate either dangers or channels. 

Those placed by the United States Coast Survey are so 
colored and numbered that in entering a bay, harbor, or chan- 
nel, red buoys with even numbers shall be passed on the star- 
board or right hand, black buoys with odd numbers on the 
port hand or left hand, and buoys with red and black stripes, 
on either hand. Buoys in channel-ways are colored with 
alternate white and black vertical stripes. 

On dangerous coasts, self-ringing bells and " fog-whistles " 
are used. 



TIDE WATERS. 



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144* LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 







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THE CHART. 



145 



CHAPTER YI. 



THE CHART. 



(229.) Having determined the lines of high and low water, 
the position of the channels, rocks, shoals, etc., and the 
soundings, a chart must be made, on which all these are laid 
down in their proper places. For scales see Art. (171.) 

The high-water line is platted like fig. 172 

the bounding lines of a farm. The 
points determined in the low-water 
line, and the positions of the boat, de- 
termined by the method given in Art. 
(213), are fixed on the chart by one 
of the methods given in Arts. (215), , 
(216), and (217). Contour curves are 
drawn as in land topography (Part 
IY.), for the first four fathoms. These 

may be indicated by dotted lines, as in Fig. 172, or they may 
be shaded with Indian-ink, as in Fig. 173. 

Fig. 173. 




V 

* Us 




mmmmmm 



Beyond four fathoms, the depths are noted in fathoms and 
vulgar fractions. 



146 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(230.) Various conventional signs are used; some of the 
principal ones are given in Figs. 174-194 : 



Fig. 174. 
Rocky shore. 



Fig. 175. Fig. 176. 

Rocks always bare. Low, swampy shore. 




Fig. 177. 



Fig. 178. =S 

Rocks sometimes 
bare. 



Fig. 179. 
Sandy shore, with hillocks. 




Fig. 180. 
:' SAND ISLAND 5 



j§£s0^ 



fF/SH WEirs 



Fig. 182. 
(SAND ALWAYS COVERD ) 



Fig. 183. 
£sand sometimes bare it 



Anchorage for ships, -r 186 

Anchorage for coasters, I 187 



Fig. 184. 


Fig. 


185. 




ECTION OF THE CURRENT 


«^'/vo 








^CURRENT 




Buoys. 

i\ 189 


Light-house, 




#192 


Wrecks. 

tr 19 ° 


Signal-house, 




i A 193 


Harbors. 









Rocks always covered, T 



188 



t& 



191 



Channel-marks g^ O 194 



PART VIII. 
SPHERICAL SURVEYING, OR GEODESY. 



CHAPTEE I. 



THE FIELD-WORK. 



(231.) Nature. It comprises the methods of surveying sur- 
faces of such extent that the curvature of the earth cannot be 
neglected. The method of triangulation is usually employed. 

(232.) Triangular Surveying is founded on the method 
of determining the position of a point by the intersection of 
two known lines. Thus, the point P 
is determined by knowing the length 
of the line A B, and the angles P B A A V P 

and P A B, which the lines P A and 
P B make with A B. By an extension / 

of the principle, a field, a farm, or a ^ /_ \b 

country, can be surveyed by measuring 

only one line, and calculating all the other desired distances, 
which are made sides of a connected series of imaginary Tri- 
angles, whose angles are carefully measured. The district 
surveyed is covered with a sort of net- work of such triangles, 
whence the name given to this kind of surveying. It is more 
commonly called " Trigonometrical Surveying," and some- 
times " Geodesic Surveying," but improperly, since it does 
not necessarily take into account the curvature of the earth, 
though always adopted in the great surveys in which that is 
considered. 



148 LEVELLING, TOPOGEAPHY, AND HIGHER SURVEYING. 

(233.) Outline of Operations. A base-line, as long as possi- 
ble (five or ten miles in surveys of countries), is measured with 
extreme accuracy. 

From its extremities, angles are taken to the most distant 
objects visible, such as steeples, signals on mountain-tops, etc. 

The distances to these and between these are then calcu- 
lated by the rules of trigonometry. 

The instrument is then placed at each of these new sta- 
tions, and angles are taken from them to still more distant 
stations, the calculated lines being used as new base-lines. 

This process is repeated and extended till the whole dis- 
trict is embraced by these " primary triangles " of as large 
sides as possible. 

One side of the last triangle- is so located that its length 
can be obtained by measurement as well as by calculation, 
and the agreement of the two proves the accuracy of the 
whole work. 

"Within these primary triangles, secondary or smaller tri- 
angles are formed, to fix the position of the minor local details, 
and to serve as starting-points for common surveys with chain 
and compass, etc. Tertiary triangles may also be required. 

The larger triangles are first formed, and the smaller ones 
based on them, in accordance with the important principle in 
all surveying operations, always to work from the whole to 
the parts, and from greater to less. 

- (234.) Measuring a Base. Extreme accuracy in this is neces- 
sary, because any error in it will be multiplied in the sub- 
sequent work. The ground on which it is located must be 
smooth and nearly level, and' its extremities must be in sight 
of the chief points in the neighborhood. Its point of begin- 
ning must be marked by a stone set in the ground with a bolt 
let into it. Over this a theodolite or transit is to be set, and 
the line " ranged out." The measurement may be made with 
chains (which should be formed like that of a watch), etc., but 
best with rods. "We will notice, in turn, their materials, sup- 
ports, alignement, levelling, and contact. 



THE FIELD-WORK. 149 

As to materials, iron, brass, and other metals, have been 

used, but are greatly lengthened and shortened by changes of 

temperature. Wood is affected by moisture. Glass rods and 

tubes are preferable on both these accounts. But wood is the 

most convenient. Wooden rods should be straight-grained 

white pine, etc., well seasoned, baked, soaked in boiling oil, 

painted and varnished. They may be trussed, or framed like 

a mason's plumb-line level, to prevent their bending. Ten or 

fifteen feet is a convenient length. Three are required, which 

may be of different colors, to prevent mistakes in recording. 

They must be very carefully compared with a standard measure. 

^Supports must be provided for the rods, in accurate work. 

Posts, set in line at distances equal to the length of the rods, 

may be driven or sawed to a uniform line, and the rods laid 

on them, either directly or on beams a little shorter. Tripods, 

or trestles, with screws in their tops to raise or lower the ends 

of the rods resting on them, or blocks with three long screws 

passing through them and serving as legs, may also be used. 

Staves, or legs, for the rods have been used, these legs bearing 

pieces which can slide up and down them, and on which the 

rods themselves rest. 

The alignement of the rods can be effected, if they are laid 
on the ground, by strings, two or three hundred feet long, 
stretched between the stakes set in the line, a notched peg be- 
ing driven when the measurement has reached the end of one 
string, which is then taken on to the next pair of stakes ; or, 
if the rods rest on supports, by projecting points on the rods 
being aligned by the instrument. 

The levelling of the rods can be performed with a common 
mason's level ; or their angle measured, if not horizontal, by 
a " slope-level." 

The contacts of the rods may be effected by bringing them 
end to end. The third rod must be applied to the second be- 
fore the first has been removed, to detect any movement. The 
ends must be protected by metal, and should be rounded' (with 
radius equal to length of rod), so as to touch in only one point. 
Kound-headed nails will answer tolerably. Better are smajl 



150 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

steel cylinders, . horizontal on one end and vertical on the 
other. Sliding ends, with verniers, have been used. If one 
rod be higher than the next one, one must be brought to touch 
a plumb-line which touches the other, and its thickness be 
added. To prevent a shock from contact, the rods may be 
brought not quite in contact, and a wedge be let down be- 
tween them till it touches both at known points on its gradu- 
ated edges. The rods may be laid side by side, and lines 
drawn across the end of each be made to coincide or form one 
line. This is more accurate. Still better is a " visual con- 
tact," a double microscope with cross-hairs being used, so 
placed that one tube bisects a dot at the end of one rod, and 
the other tube bisects a dot at the end of the next rod. The 
rods thus never touch. The distance between the two sets of 
cross-hairs is of course to be added. 

A base could be measured over very uneven ground, or 
even water, by suspending a series of rods from a stretched 
rope by rings in which they can move, and levelling them and 
bringing them into contact as above. 

The most perfect base-measuring apparatus is that used on 
the United States Coast Survey. 1 It consists of a bar of brass 
and a bar of iron, a little less than six metres long, supported 
parallel to each other, firmly attached to a block at one end, 
and left free to move at the other, so that the entire contrac- 
tion and expansion are at that end. At right angles to these 
bars is a short lever, called the " lever of compensation." It 
is attached to the lower (brass) bar at the free end by a hinge, 
and an agate knife-edge on the lever rests against a steel plate 
at the end of the iron bar. 

When the temperature is raised, both bars expand, but the 
brass one more than the iron one, so that the upper end of the 
lever of compensation is thrown back. A knife-edge, turned 
outward, is placed on the lever, at such a distance from the 
other knife-edge and the hinge, that it shall remain unmoved 
by equal changes of temperature in the two bars. 

Brass and iron, exposed to the same temperature, will not 

1 For a full description, see Coast Survey Report of 1854. 



THE FIELD- WOKK. 151 

heat equally in equal times. To overcome this difficulty, the bars 
are given equal absorbing surfaces, but their cross-sections are 
adapted to their different specific heats and conducting powers. 

The knife-edge on the upper end of the lever of compen- 
sation presses against a short sliding rod, supported on the upper 
(iron) bar, and held firmly against the lever by a spiral spring. 
The sliding rod is terminated on the outer end by an agate plane. 

The end of the apparatus we have been considering is 
called the compensating end. "We will now consider the sec- 
tor end, where are arranged the parts for adjusting the con- 
tacts between the successive rods in measuring ; and for de- 
termining the inclination of the rod on sloping ground. 

This end also terminates in a sliding rod, bearing on its 
extremity an agate knife-edge, placed horizontally, and resting 
by its inner end against an upright " lever of contact." This 
lever is fastened by a hinge at the lower end, and its upper 
end rests against a tongue, attached to the " level of contact," 
which is mounted on trunnions. When the sliding rod is 
moved in, the lever of contact presses against the tongue of 
the level of contact and turns the level. The inner end of the 
level-tube is weighted so as to insure a constant pressure when 
the contact is made between two rods, and the bubble is 
brought to the centre. The sector is an arrangement for de- 
termining the angle at which the rod is inclined. 

The whole apparatus is enclosed in a double tin tubular 
case, only the ends of the sliding rods, bearing the agates, 
being exposed. The observations are taken through glass 
doors in the side of the tube. The extreme length is six me- 
tres. Two of these tubes are used in measuring a base, and 
each is supported by two trestles. The tubes are aligned by 
the aid of a transit. 

On one base, seven miles long, measured with this appara- 
tus, the greatest supposable error was computed, from remeas- 
urements, to be less than six-tenths of an inch. On another 
base, six and three-quarter miles long, the probable error was 
less than one-tenth of an inch, and the greatest supposable 
error was less than three-tenths of an inch. 



152 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

(235.) Corrections of Base. If the rods were not level, their 
length must be reduced to its horizontal projection. This 
would be the square root of the difference of the squares of 
the length of the rod (or of the base), and of the height of one 
end above the other ; or the product of the same length by 
the cosine of the angle which it makes with the horizon. 1 

If the rods were metallic, they would need to be corrected 

of its 



for temperature. Thus, if an iron bar expands 100 Zo(n r 
length for 1° Fahr., and had been tested at 32°, and a base 
had been measured at 72° with such a bar 10 feet long, and 
found to contain 3,000 of them, its apparent length would be 
30,000 feet, but its real length, would be 8.4 feet more. 

EXPANSION FOR 1° FAHRENHEIT. 

Brass bar = 0.00001050903; 

Iron bar == 0.000006963535 ; 
Platinum = 0.0000051311; 
Glass = 0.0000013119 ; 
White Pine = 0.0000022685. 



(236.) Reducing the Base to the Level of the Sea, Let 



Fig. 196. 




A B = a be the measured base, and 
A' B / = x, the base reduced to the 
level of the sea, h the height of the 
measured base above the level of the 
sea, and r the radius of the earth to 
the level of the sea. Then we have : 
r + h : r :: a : x 



x = a 



ah 



a—x 



r + h 

1+T 



r + h 
ah 

r _ah( h \~ x 
K ' ~ r V r J ' 



1 More precisely, A being this angle, and not more than 2° or 3°, the difference 
between the inclined and horizontal lengths equals the inclined or real length mul- 
tiplied by the square of the minutes in A, and that by the decimal 0.00000004231. 



THS FIELD-WORK. 153 

Developing by the binomial formula, we get : 

h V h 3 

a — x = a—— a—i-\-a— % —, etc. 

As h is very small in comparison with r, the first term of the 
correction is generally sufficient. 

(237.) A Broken Base. "When the angle C is very obtuse, 
the lines A C and C B being measured, and forming nearly a 

Fig. 197. 



c 



straight line, the length of the line A B is found thus : Naming 
the lines, as is usual in trigonometry, by small letters cor- 
responding to the capital letters at the angles to which they 
are opposite, and letting K = the number of minutes in the 
supplement of the angle C, we shall have : 

7 T7"2 

A~B = c = a + b- 0.000000042308 x a 



a + b 
Log. 0.000000042308 = 2.6264222 - 10. 

(238.) Base of Verification. As mentioned in Art. (233), a 
side of the last triangle is so located that it can be measured, 
as was the first base. If the measured and calculated lengths 
agree, this proves the accuracy of all the previous work of 
measurement and calculation, since the whole is a chain of 
which this is the last link, and any error in any previous part 
would affect the very last line, except by some improbable 
compensation. How near the agreement should be, will de- 
pend on the nicety desired and attained in the previous opera- 
tions. Two bases, 60 miles distant, differed on one great 
English survey 28 inches ; on another, 1 inch ; and on a French 



154 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

triangulation extending over 500 miles, the difference was less 
than 2 feet. Results of equal or greater accuracy are obtained 
on the United States Coast Survey. " The Fire Island base, 
on the south side of Long Island, and the Kent Island base in 
Chesapeake Bay, are connected by a primary triangulation. 
This Kent Island base is 5 miles and 4 tenths long, and the 
original Fire Island base is 8 miles and 7 tenths. The short- 
est distance between them is 208 miles, but the distance 
through the triangulation is 320. The number of intervening 
triangles is 32, yet the computed and measured lengths of the 
Kent Island base exhibit a discrepancy no greater than 4 
inches.". 

(239.) Choice of Stations. The stations, or " trigonometri- 
cal points," which are to form the vertices of the triangles, 
and to be observed to and from, must be so selected that 
the resulting triangles may be " well-conditioned," i. e., may 
have such sides and angles that a small error in any of the 
measured quantities will cause the least possible errors in the 
quantities calculated from them. The higher calculus shows 
that the triangles should be as nearly equilateral as possible. 
This is seldom attainable, but no angle should be admitted 
less than 30°, or more than 120°. When two angles only 
are observed, as is often the case in the secondary triangu- 
lation, the unobserved angle ought to be nearly a right 
angle. 

To extend the triangulation , by continually increasing the 
sides of the triangles, without introducing " ill-conditioned " 
triangles, may be effected as in Fig. 198. A B is the measured 
base, C and D are the nearest stations. In the triangles ABC 
and A B D, all the angles being observed, and the side A B 
known,- the other sides can be readily calculated. Then, in 
each of the triangles D A C and D B C, two sides and the 
contained angles are given to find D C, one calculation check- 
ing the other. DC then becomes a base to calculate EF, 
which is then used to find G H, and so on. 

The fewer primary stations used the better, both to pre- 



THE FIELD-WORK. 



155 



vent confusion and because the smaller number of triangles 
makes the correctness of the results more " probable." 

Fig. 198. 




The United States Coast Survey, under the superintend- 
ence of Prof. A. D. Baehe, displays some fine illustrations of 
these principles, and of the modifications they may undergo 
to suit various localities. The figure* on the next page repre- 
sents part of the scheme of the primary triangulation resting 
on the Massachusetts base, and including some remarkably 
well-conditioned triangles, as well as the system of quadri- 
laterals, which is a valuable feature of the scheme when the 
sides of the triangles are extended to considerable lengths, 
and quadrilaterals, with both diagonals determined, take the 
place of simple triangles. 

The engraving is on a scale of 1 : 1200,000. 



156 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 
Fiff. 199. 



/ 




Boston 



hi 

o 




THE FIELD-WORK. 



157 



Fig. 200. 



(240.) Signals. They must be high, conspicuous, and so 
made that the instrument can be placed precisely under them. 

Three or four timbers framed into a 
pyramid, as in Fig. 200, with a long mast 
projecting above, fulfil the first and last 
conditions. The mast may be made ver- 
tical by directing two theodolites to it, and 
adjusting it so that their telescopes follow 
it up and down, their lines of sight being 
at right angles to each other. Guy-ropes 
irray be used to keep it vertical. 

A very excellent signal, used on the 
Massachusetts State Survey, by Mr. Borden, is represented in 
the three following figures. It consists merely of three stout 




Fig. 201. 



Fig. 202. 



Fig. 203. 






sticks, which form a tripod, framed with the signal- staff, by a 
bolt passing through their ends and its middle. Fig. 201 
represents the signal as framed on the ground ; Fig. 202 shows 
it erected and ready for . observation, its base being steadied 
with stones ; and Fig. 203 shows it with the staff 
turned aside, to make room for the theodolite and 
its protecting tent. The heights of these signals 
varied between 15 and 80 feet. 

Another good bignal consists of a stout post let 
into the ground, with a mast fastened to it by a 
bolt below and a collar above. By opening the 
collar, the mast can be turned down and the the- 
odolite set exactly under the former summit of the 
signal, i. e., in its vertical axis. 




158 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

A tripod of gas-pipe has been used to support the signal in 
positions exposed to the sea, as on shoals. It is taken to the 
desired spot in pieces, and there screwed together and set up. 

Signals should have a height equal to at least 7 *- Q of their 
distance, so as to subtend an angle of half a minute, which 
experience has shown to be the least allowable. 

To make the tops of the signal-masts conspicuous, flags 
may be attached to them ; white and red, if to be seen against 
the ground, and red and green if to be seen against the sky. 1 
The motion of flags renders them visible, when much larger 
motionless objects are not. But they are useless in calm 
weather, A disk of sheet-iron, with a hole in it, is very con- 
spicuous. It should be arranged so as to be turned to face 
each station. A barrel, formed of muslin sewed together, four 
or live feet long, with two hoops in it two feet apart, and its 
loose ends sewed to the signal-staff, which passes through it, is 
a cheap and good arrangement. A tuft of pine-boughs fast- 
ened to the top of the staff, will be well seen against the sky. 

In .sunshine a number of pieces of tin, nailed to the staff at 
different angles, will be very conspicuous. A truncated cone 
of burnished tin will reflect the sun's rays to the eye in almost 
every situation. 

The most perfect arrangement is the "heliotrope," in- 
vented by Gauss. This consists of a mirror a few inches 
square, so mounted on a telescope, near the eye-end, that the 
reflection of the sun may be thrown in any desired direction. 
They have been observed on at a distance of 80 or 90 miles, 
when the outlines of the mountains on which they were placed 
were invisible. A man, called a " heliotroper," is stationed 

1 To determine at a 'station A, 
whether its signal can be seen from 



B, projected against the sky or not, 
measure the vertical angles B A Z 
and Z A C. If their sum equals or 
exceeds 180°, A will be thus seen 
from B. If not, the signal at A 
must be raised till this sum equals 
180°. 




THE FIELD-WORK. 



159 



at the instrument. He directs the telescope toward the sta- 
tion at which the transit is placed for observation, and keeps 
the mirror turned so as to reflect the sun in a direction parallel 
to the axis of the instrument. This he accomplishes by caus- 
ing the reflection to pass through two perforated disks, mounted 
on the telescope, one near the object-end, and the other near 
the mirror. 

For night-signals, an Argand lamp is used ; or, best of 
all, Drummond's light, produced by a stream of oxygen gas 
directed through a flame of alcohol upon a ball of lime* Its 
distinctness is exceedingly increased by a parabolic reflector 
behind it, or a lens in front of it. Such alight was brilliantly 
visible at 66 miles' distance. 

(241.) Observations of the Angles. These should be repeated 
as often as possible. In extended surveys, three sets, of ten 
each, are recommended. They should be taken on different 
parts of the circle. In ordinary surveys, it is well to employ 
the method of "traversing," Art. (94). In long sights, the 
state of the atmosphere has a very remarkable effect on both 
the visibility of the signals and on the correctness of the ob- 
servations. 

"When many angles are taken from one station, it is im- 
portant to record them by some uniform system. The form 
given below is convenient. It will be noticed that only the 
minutes and seconds of the second vernier are employed,, the 
degrees being all taken from the first : 



Observations at 



Station 
observed to. 


READINGS. 


Mean 
Eeading. 


Right or Left 
of Preced- 
ing Object. 


Remarks. 


Vernier A. 


Vernier B. 


A 
B 

C 


70° 19' 0" 
103° 32' 20" 
115° 14' 20" 


18' 40" 
32' 40" 
14' 50" 


70° 18' 50" 
103° 32' 30' r 
115° 14' 35" 


R. 
R. 





11 



160 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

When the angles are " repeated," the multiple arcs will 
be registered nnder each other, and the mean of the seconds 
shown by all the verniers at the first and last readings be 
adopted. 

The great theodolite, nsed on the Coast Survey for the 
observation of the angles in the primary triangles, has a 
horizontal circle thirty inches in diameter, graduated to ^.Ye 
minutes, and reading to single seconds by three micrometer 
microscopes, placed 120° apart. The telescope has a focal 
length of four feet. 

When the country over which the triangulation extends is 
flat, it has been found necessary to elevate the transit some 
distance from the surface of the ground, the stratum of air 
near the surface being so disturbed by exhalations and ine- 
qualities of temperature and density as to render accurate 
observation impossible. The plan adopted on the Coast Sur- 
vey is as follows : On the top of a signal-tripod, forty-three 
feet high, is placed a cap-block, into which is mortised a square 
bole to receive the signal-pole. Around the tripod, but not 
touching it, is erected a rectangular scaffold, forty feet high. 
On the top of it is a platform, from which the observations 
are taken, the signal-pole being removed from the cap-block, 
and the transit placed so that its centre shall be precisely over 
the station-point. 

(242.) Reduction to the Centre. It is often impossible to 

set the instrument precisely at or under the signal which has 

Fig. 206. been observed. In such cases 

--^^l proceed thus : Let C be the 

— \^^^ centre of the signal, and R C L 

x^j/^ the desired angle, R being the 

D V< C^^***<^ right-hand object and L the 

^~^~^^^^ R left-hand one. Set the instru- 
ment at D, as near as possible 
to C, and measure the angle KDL. It may be less than 
R C L, or greater than it, or equal to it, according as D lies 
without the circle passing through C, L, and R, or within it, 



THE FIELD-WORK. IQ± 

or in its circumference. The instrument should be set as 
nearly as possible in this last position. To find the proper 
correction for the observed angle, observe also the angle LDC 
(called the angle of direction), counting it from 0° to 360°, 
going from the left-hand object toward the left, and measure 
the distance D C. Calculate the distances C E and C L with 
the angle E D L, instead of E C L, since they are sufficiently 
nearly equal. Then, 

CD.sin.(EDL +LDC) C D . s in. L DC 
KLL-KDL+ CR7^n7P " C L . sin. 1"~ 

The last two terms will be the number of seconds to be 
added or subtracted. The trigonometrical signs of the sines 
must be attended to. The log. sin. 1" = 4.6855749. Instead 
of dividing by sin. 1", the correction without it, which will be 
a very small fraction', may be reduced to seconds by multiply- 
ing it by 206265. 

Example.— Let E D L = 32° 20 A 18".06 ;LDC = 101° 15' 
32".4; CD = 0.9; C E =± 35845.12 ; CL = 29783.1. 

The first term of the correction will be + 3 /r .Y50, and the 
second term — 6 /r .113. Therefore, the observed angle E D L 
must be diminished by 2". 363, to reduce it to the desired an- 
gle ECL. 

Much calculation may be saved by taking the station D so 
that all the signals to be observed can be seen from it. Then 
only a single distance and angle of direction need be measured. 

It may also happen that the centre, C, of the 
signal cannot be seen from D. Thus, if the sig- 
nal be a solid circular tower, set the theodolite 
at D, and turn its telescope so that its line of 
sight becomes tangent to the tower at T, T' ; T ^ 
measure on these tangents equal distances, D E, 
D F, and direct the telescope to the middle, G, 
of the line E F. It will then point to the centre, d 

C ; and the distance D C will equal the distance from D 
to the tower plus the radius obtained by measuring the cir : 
cumference. 





162 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

If the signal be rectangular, measure D E, D T. Take 
fig 208 an y P omt G on D E, and on D F set off D H 
DF 
= D G jyv • Then is G H parallel to E F (since 

D G : D H : : D E : D F), and the telescope di- 
rected to its middle, K, will point to the middle 
of the diagonal E F. We shall also have D C 

= I ,K§|. 

Any such case may be solved by similar methods. 

The "j?hase " of objects is the effect produced by the sun 
shining on only one side of them, so that the telescope will be 
directed from a distant station to the middle of that bright 
side instead of to the true centre. It is a source of error to 
be guarded against. Its effect may, however, be calculated. 

When the signal is a tin cone : 

Let r = radius of the signal, 

Z = angle at the point of observation between the 

sun and the signal, 
D = the distance. 

_. _ . ^cos. 2 -|Z 

I hen, the correction = ± ^ — -. — ^ . 

I) sm. 1 

(243.) The Angles. The triangles observed are supposed to 
have sides of such length that the sum of the three angles ex- 
ceeds 180° by a certain sensible quantity called the " spherical 
excess." This is usually only a few seconds. For a triangle 
containing about 76 square miles, which, if equilateral, would 
have sides 13 miles long, the spherical excess is only one sec- 
ond. For a triangle with sides of 102 miles it is one minute. 

It must be determined before we can know how much the 
error is, and therefore what the correct sum and correction 
should be. 

(244.) The true spherical excess is found by this principle : 
" The surface of a spherical triangle is measured by the excess 



THlJ FIELD-WOEK. 1^3 

of its angles above two right angles multiplied by the trirec- 
t angular triangle." 1 

Hence the surfaces of spherical triangles are to each other 
as their respective spherical excesses. 

Let s = surface of given triangle, 

t = surface of trirectangular triangle, 
e = spherical excess of given triangle, 
e f = spherical excess of trirectangular triangle. 
Then, we have : 

s : t : : e : e' . 

t — \ surface of sphere = \ x 4 ixr 1 — \ nr*. 
e f = (3 x 90°) - 180° = 90°. 

Then, s : -i-Trr 3 : : e : 90°. 
Whence, e = s x - a — in seconds. 

s and r are in the same unit of measure. 

The fraction is a constant quantity whose logarithm is 
10.6746069, the mean radius of the earth being taken as 
20888629 feet ; the greater radius being 20923596, and the 
smaller radius 20853662. 2 

The surface s, being very small compared with r 2 , may be 
obtained with sufficient accuracy for this object by treating 
the triangle as if it were plane. 

Then, when two sides and the contained angle are given, 
we have : 

s — \a~b . sin. C. 

When two angles and the included side are given, we have : 

2 sin. B . sin. C 
S=r ^ a X sin. (B + C) * • 

Approximately, the spherical excess (in seconds) equals 
the area (in square miles) divided by 75.5. 

1 Davies's I egendre, Book IX., prop. 18. 2 According to Sir John Herschel. 



164 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Having found the spherical excess, if the sum of the angles 
of the triangle does not equal 180° plus this excess, the differ- 
ence is distributed among them as in Art. (245). 

(245.) Correction of the Angles. When all the angles of any 
triangle can be observed, their sum should equal 180° plus the 
" spherical excess." If not, they must be corrected. If all 
the observations are considered equally accurate, one-third of 
the difference of their sum from 180° plus the spherical excess, 
is to be added to or subtracted from each of them. But if the 
angles are the means of unequal numbers of observations, their 
errors may be considered to be inversely as those numbers, 
and they may be corrected by this proportion : As the sum of 
the reciprocals of each of the three numbers of observations is 
to the whole error, so is the reciprocal of the number of obser- 
vations of one of the angles to its correction. 

It is still more accurate, but laborious, to apportion the 
total error, or difference from 180° plus the spherical excess, 
among the angles inversely as the " weights." x On the United 
States Coast Survey, in six triangles measured in 1844 by 
Prof. Bache, the greatest error was six-tenths of a second. 

(248.) Interior Filling-up. The stations whose positions 
have been determined by the triangulation are so many fixed 
points, from which more minute surveys may start and inter- 
polate any other points. The trigonometrical points are like 
the -observed latitudes and longitudes which the mariner ob- 
tains at every opportunity, so as to take a new departure from 
them, and determine his course in the intervals by the less pre- 
cise methods of his compass and log. The chief interior points 
may be obtained by " secondary triangulation," and the minor 
details be then filled in by any of the methods of surveying, 
with chain, compass, or transit, already explained, or by the 
plane-table. 

With the transit or theodolite, " traversing " is the best 
mode of surveying, the instrument being set at zero, and being 

1 L. S., Art. (369). 



CALCULATING THE SIDES OF THE TRIANGLES. 



165 



then directed from one of the trigonometrical points to 
another, which line therefore becomes the " Meridian " of that 
survey. On reaching this second point, in the course of the 
survey, and sighting back to the first, the reading should of 
course be 0°. 



CHAPTEK II. 



CALCULATING THE SIDES OF THE TRIANGLES 



(247.) One side of a spherical triangle having been meas- 
ured or calculated, and all the angles observed, the other sides 
can be computed by employing the principles of spherical 
trigonometry. This, however, is very laborious, and other 
methods have been adopted which, with less work, give results 
equally accurate. 

(248.) Delambre's Method. 

Imagine the three angular 
points of each spherical tri- 
angle to be joined by straight 
lines, chords of the arc, so as to 
form a plane triangle, as in 
Fig. 209. Eeduce the given 
curved side to its chord, and 
the spherical angles to the 
plane angles contained by these 
chords. Compute the other 
sides or chords by plane trigonometry, and then calculate the 
arcs corresponding to them. 

To reduce any arc to its chord we have : 

Chord of an arc a = 2 sin. \ a. 
Or, if a be the arc in terms of the radius : 
Chord of a = a — -fa a 3 . 




166 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

To reduce an angle of a spherical triangle to the corre- 
sponding angle between the chords of the including arcs : 

Let ABC, Fig. 210, be a spherical triangle, and O the 

centre of the sphere. It is required to reduce the spherical 

FiQ 210 angle at A to the plane angle between 

A the chords A B and A C. Draw O G 

and O H parallel to A B and A C, and 

\\ prolong the arcs to Gr and H. The 

'' \v lines O D and O E, bisecting the chords 

\ A B and A C, will be perpendicular to 

1 them, and also to O Gr and O H. Then 

/ u \ DGandEH are quadrants. Now, in 

L--'' \J the spherical triangle A G H, having 

■ the arcs A G and A H, and the included 

angle G A H, the measure of the angle G O H is found by 
spherical trigonometry. But G O H = B A C, the required 
angle. 

The sum of the three plane angles thus found will be 
equal to two right angles, if the observations of the spherical 
angles and the work of reducing have been correctly done. 

(249.) Legendre's Method. His theorem is this : " In any 
spherical triangle, the sides of which are very small compared 
to the radius of the sphere, if each of the angles be diminished 
by •£■ of the true spherical excess, the sines of these angles will 
be proportioned to the lengths of the opposite sides ; and the 
triangle may therefore be calculated as if it were a plane one." 

This is the easiest method. 

All three methods were used for the French " Base du 
systeme metrique." 

In the British "Ordnance Survey" the triangles were 
mostly calculated by the second method, and checked by the 
third. 

The difference on 100 miles is only a fraction of a yard. 

(250.) Co-ordinates of the Points. The polar spherical co- 
ordinates of a point with respect to another point are these : 



CALCULATING THE SIDES OF THE TRIANGLES. 



167 



the length of the arc of the great circle passing through the 
points, and its azimuth, i.e., the angle it makes with the 
meridian passing through one of its points. 

The rectangular spherical coordinates of a point have for 
axes the meridian passing through the origin, and a per- 
pendicular to it. For short distances these may be regarded 
as in one plane. For greater distances new meridians must 
be taken — say, not farther apart than 50 miles. 

"Within that limit the successive triangles may be con- 
ceived to be turned down into the same plane. 

The astronomical coordinates of a point are its latitude 
and longitude. These are determined by practical astronomy. 

The transformation of these coordinates to polar or rec- 
tangular, and vice versa, is very important. It is done by 
spherical trigonometry. The latitude and longitude of any 
one point are very accurately determined by the mean of a 
great number of astronomical observations, and those of the 
other points are calculated from these. Thos,e of some other 
points may be observed as checks. 

It is found that the observed and calculated latitude and 
longitude of a place do not always agree, even when the earth 
is considered as an ellipsoid of revo- 
lution ; in consequence of the irreg- 
ularity of the form of the earth; The 
difference of the "geodesic" from 
the astronomical determination of 
difference of latitude and longitude, 
is called the " station error." 

A " geodesic line " is the short- 
est line which can be drawn on the 
ellipsoid, corresponding to an arc 
of a great circle on the sphere. It 
is the line of least curvature. 



(251.) Pkob. 1.. Given latitude ae = latitude. 

and longitude of A, and the azimuth and distance' from A to B. 
Required the latitude and longitude of B, and the azimuth 




(Lv\gJ 



168 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 



from B to A. The distance is measured on the arc of a great 
circle passing through those points, the earth being assumed 
to be a sphere. 

We have given two sides and the included angle, to find 
the remaining parts. 

By spherical trigonometry we have : 



tan. J(B + P) = cot.-|A 



cos.j(AP-AB) 

cos.i(AP + AB) 



i/T3 -dn l1A Bin,i(AP-AB) 

tan.KB-P)^cot.iA sin>i ; Ap + X B j. 

The azimuth from B to A = B = £(B + P) -f i(B - P). 
The difference of long. = P = J(B + P) - J(B - P). 
To find the co-latitude of B = P B, we have : 

tan . t P B = W(AP-AB)riM(B + P) 

sm. i( B — P ) 



Fig. 212. 




(252.) Otherwise. — Let fall from B, 
B perpendicular to A P. Then, 

tang. AC= tang. A B . cos. P A B. 

PC = PA-AC. 

^ ^ cos. A B . cos. P C 







cos. A C. 


sin 


.A. 


sin. A B 




sin. 


,PB * 


sin. 


A. 


sin. A P 



sin. A B P — 

sm. P B 

(253.) Peob. 2. Given latitude and longitude of A and B 



CALCULATING THE SIDES OF THE TRIANGLES. 



169 



to find the distance between 
them and the azimuth from 
each to the other ; i. e., to find 
the length and direction of the 
arc of a great circle passing 
through those points. 

The angle of P is the differ- 
ence of longitude, P B is the 
co-latitude of B, and P A is the 
co-latitude of A. Then we have 
two sides and the included 
angle to find the remaining 
parts. 



tan.J(B + A) = cot. -£P 



Fig. 213. 




COS. j(PA-PB) 

cos. i(PA + PB)' 



tan. i(B 



tan 



A^-cot , P ™-*(rA-PB) 
A)_cot.^P s m^ (pA + pB) . 

B=|-(B + A)+f(B-A). 
A = i(B + A)-MB-A). 

i a B = tan -i(PA-PB)sin.i(B + A) 
2 ' sin. \ ( B — A) 



This is strictly a case of spherical location, required in 
planning a road between two distant points, and in navigating 
a vessel. 

The distance may also be found thus : 

Let a and j3 represent the co-latitudes of A and B. 

cos. AB = cos. a . cos. (3 + sin. a . sin. (5 . cos. P. 

Put tang. <f> = tan. a . cos. P ; 

cos. a . cos. (13 —0) 



Then, cos. A B 



cos. <p 



170 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

Any other sets of three parts of the triangle PAB being 
given, the rest can be found by spherical trigonometry. 

(254.) For great accuracy, the earth must be regarded as a 
spheroid. The following formulas for computing the geodesic 
latitudes, longitudes, and azimuths of points of a triangula- 
tion, are from Captain Lee's Tables and Formulas : 

Let K = distance in yards between two stations, the lat- 
itude and longitude of one of which are known, and u" this 
same distance converted to second of arc. 

L = latitude of first station. 

M = longitude of first, + if west. 

Z = azimuth of second station at first, counted from the 
south around, by the west, from 0° to 360°. The algebraic 
signs of the sine and cosine of this angle must be carefully 
attended to. 

L', M', Z', the same things at second station, or quantities 
required. 

a = the equatorial radius. 

e = the eccentricity = 0.0817 = \ (— — % — J- 

E = the radius of curvature of the meridian, in yards. 
N = the radius of curvature of a section perpendicular to 
the meridian, in yards. 

K K(l-Vsin. 2 L)* 



" ~~ N sin. 1" ~ a sin. 1" 

L'=.L-(1 + J cos. 2 L)u" cos. Z- (1 + e 2 cos. a L) 
(^"sin.'Z) a tan. L x i sin. 1". 

,, u" sin. Z 
cos. L 

Z' = 180° + Z - "l'-™^ sin. i ( L + I/), or 

COS. J-J' 

Z r = 180° + Z - ( u" sin. Z tan. L + u" 2 sin. Z cos. Z \ sin. 1"). 



CALCULATING THE SIDES OF THE TRIANGLE. 



171 



,™ . u ff sin. Z . „ ,_ T# . ,^ r , ,,-. . 

The quantity jt- sin. £ ( L + I/), or ( M'— M ) sin. J 

(L + I/), by which the azimuth at one end of a line exceeds 
the azimuth at the other, is called the convergence of the 
meridians. 

In terms of the coordinates of rectangular axes referred to 
one of the points of the triangulation, the latitude and longi- 
tude of which are-known, y being the ordinate in the direction 
of the meridian, and x the ordinate perpendicular to it : 

L'=L± ^ y .„ -4-sin. 1" L T ? 1/y ) . tan. (l± p ? ^A 
E sm. 1" 3 V3ST sin. 1/ V Rsin. rj 

\JN sm. IV 



cos. I/' 



Z'= 270° ± 



N sin. 1 



7, tan. I/. 



(255.) Calculation of a spherical triangle by Legendre's 
method. (Art. 249.) 

The following example is from the United States Coast 
Survey : 



No. 


Denomi- 
nation. 


Observed 
angles. 


Correc- 
tion. 


Spherical 
angles. 


Spherical 

excess. 


Plane, angles, 
and distances. 


Loga- 
rithms. 


1 


Prince. . 
Buck... 
Hill.... 


41 47 41.79 
81 13 13.78 
56 59 07.39 

Buck t 


-0.60 
-0.60 
-0.60 

o Hill. . . 


41.19 
13.18 
06.79 


0.39 
0.39 
0.38 


41 47 40*80 

81 13 12.79 

56 59 06.41 

m. 

19189.80 

28456.10 

24144.18 


0.1762239 
9.9948811 
9.9235180 

4.2830705 




Prince 
Prince 


to Hill . . 




4.4541755 




to Buck. 




4.3828124 



The data for calculation are one side (Buck to Hill), and 
the observed angles. 

To determine the spherical excess, apply the formula given 

in Art. 244 : 

648000" 
e = s X — . 

As the surface, s, of the triangle is very small, compared 



172 LEVELLING, TOPOGRAPHY, AND HIGHER SURVEYING. 

with the diameter of the earth, it may be obtained with suf- 
ficient accuracy, for this purpose, by treating the triangle as 
if it were plane. Then, the three angles and one side being 
given, we have the formula : 

, 2 Sin. B. Sin. C 
S = * a Sin. A. ? 

in which a is the given side ; B and C, the adjacent angles, 
and A, the opposite angle. 

In getting the value of the fraction in the formula for the 
spherical excess, the radius of the earth, r, must be taken in 
the same unit of measure as s. The values used on the Coast 
Survey are: Equatorial radius, 6377397.16 metres; polar ra- 
dius, 6356078.96 metres ; and mean radius, 6366738.06 metres. 

To find Log. s. 

Log. \ = 1.6989700 

Log. a* (19189.80) 2 = 8.5661410 

o / // 

Log. sin. B (81 13 13.78) = 9.9948813 
Log. sin. C (56 59 07.39) = 9.9235193 
Co-log. sin. A (41 47 41.79) = 0.1762216 



To find Lo 



Log. *. = 8.3597332 

648000" 



Log. 648000' = 5.8115750 

Co-log. 7T (3.1415927) = 9.5028501 
Co-log. r* (6366738.06) 2 = 6.3921660 



Log. 6 ^' =9.7065911 

& 7r r 

Log. s. = 8.3597332 

Log. e, spherical excess = -.0663243 

Spherical excess = 1 // .16 

The difference between the sum of the observed angles 
and 180° plus the spherical excess (l'M6), is 1".80, which 
will make a correction for each angle of 0".60. Placing this 



CALCULATING THE SIDES OF THE TRIANGLE. 



173 



correction in the fourth column, and subtracting it from the 
observed angles, we get the corrected spherical angles for the 
fifth column. One-third of the spherical excess (sixth column) 
is then subtracted from the spherical angles to reduce them to 
plane angles, which are placed in the seventh column. Using 
these plane angles, and the given side, and applying the sine 
proportion, we have : 



To find b. 



Log. a 



= 4.2830705 
Log. sin. B = 9.9948811 
Co-log. sin. A = 0.1762239 

Log. h = 4.4541755 

Prince to Hill = 28456.10 



To find c. 



Log. a 
Log. sin. C 
Co-log. sin. A 

Log. c 



= 4.2830705 
= 9.9235180 
= 0.1762239 

= 4.3828124 



Prince to Buck = 24144.18 



The logarithms of the sides and of the sines of the plane 
angles are placed in the last column. For convenience in cal- 
culation, the co-log. of angle opposite the given side is taken. 



THE END 



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